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<!DOCTYPE html><html lang="en-US"><head><title>Math 1. Vectors</title><meta property="og:title" content="Math 1. 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/*!
* Marp default theme.
*
* @theme default
* @author Yuki Hattori
*
* @auto-scaling true
* @size 16:9 1280px 720px
* @size 4:3 960px 720px
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.pl-smi{color:var(--color-prettylights-syntax-storage-modifier-import)}div#\:\$p>svg>foreignObject>section .pl-ent{color:var(--color-prettylights-syntax-entity-tag)}div#\:\$p>svg>foreignObject>section .pl-k{color:var(--color-prettylights-syntax-keyword)}div#\:\$p>svg>foreignObject>section .pl-pds,div#\:\$p>svg>foreignObject>section .pl-s,div#\:\$p>svg>foreignObject>section .pl-s .pl-pse .pl-s1,div#\:\$p>svg>foreignObject>section .pl-sr,div#\:\$p>svg>foreignObject>section .pl-sr .pl-cce,div#\:\$p>svg>foreignObject>section .pl-sr .pl-sra,div#\:\$p>svg>foreignObject>section .pl-sr .pl-sre{color:var(--color-prettylights-syntax-string)}div#\:\$p>svg>foreignObject>section .pl-smw,div#\:\$p>svg>foreignObject>section .pl-v{color:var(--color-prettylights-syntax-variable)}div#\:\$p>svg>foreignObject>section .pl-bu{color:var(--color-prettylights-syntax-brackethighlighter-unmatched)}div#\:\$p>svg>foreignObject>section .pl-ii{background-color:var(--color-prettylights-syntax-invalid-illegal-bg);color:var(--color-prettylights-syntax-invalid-illegal-text)}div#\:\$p>svg>foreignObject>section .pl-c2{background-color:var(--color-prettylights-syntax-carriage-return-bg);color:var(--color-prettylights-syntax-carriage-return-text)}div#\:\$p>svg>foreignObject>section .pl-sr .pl-cce{color:var(--color-prettylights-syntax-string-regexp);font-weight:700}div#\:\$p>svg>foreignObject>section .pl-ml{color:var(--color-prettylights-syntax-markup-list)}div#\:\$p>svg>foreignObject>section .pl-mh,div#\:\$p>svg>foreignObject>section .pl-mh .pl-en,div#\:\$p>svg>foreignObject>section .pl-ms{color:var(--color-prettylights-syntax-markup-heading);font-weight:700}div#\:\$p>svg>foreignObject>section .pl-mi{color:var(--color-prettylights-syntax-markup-italic);font-style:italic}div#\:\$p>svg>foreignObject>section .pl-mb{color:var(--color-prettylights-syntax-markup-bold);font-weight:700}div#\:\$p>svg>foreignObject>section .pl-md{background-color:var(--color-prettylights-syntax-markup-deleted-bg);color:var(--color-prettylights-syntax-markup-deleted-text)}div#\:\$p>svg>foreignObject>section .pl-mi1{background-color:var(--color-prettylights-syntax-markup-inserted-bg);color:var(--color-prettylights-syntax-markup-inserted-text)}div#\:\$p>svg>foreignObject>section .pl-mc{background-color:var(--color-prettylights-syntax-markup-changed-bg);color:var(--color-prettylights-syntax-markup-changed-text)}div#\:\$p>svg>foreignObject>section .pl-mi2{background-color:var(--color-prettylights-syntax-markup-ignored-bg);color:var(--color-prettylights-syntax-markup-ignored-text)}div#\:\$p>svg>foreignObject>section .pl-mdr{color:var(--color-prettylights-syntax-meta-diff-range);font-weight:700}div#\:\$p>svg>foreignObject>section .pl-ba{color:var(--color-prettylights-syntax-brackethighlighter-angle)}div#\:\$p>svg>foreignObject>section .pl-sg{color:var(--color-prettylights-syntax-sublimelinter-gutter-mark)}div#\:\$p>svg>foreignObject>section .pl-corl{color:var(--color-prettylights-syntax-constant-other-reference-link);text-decoration:underline}div#\:\$p>svg>foreignObject>section g-emoji{display:inline-block;font-family:Apple Color Emoji,Segoe UI Emoji,Segoe UI Symbol;font-size:1em;font-style:normal!important;font-weight:var(--base-text-weight-normal,400);line-height:1;min-width:1ch;vertical-align:-.075em}div#\:\$p>svg>foreignObject>section g-emoji img{height:1em;width:1em}div#\:\$p>svg>foreignObject>section .task-list-item{list-style-type:none}div#\:\$p>svg>foreignObject>section .task-list-item label{font-weight:var(--base-text-weight-normal,400)}div#\:\$p>svg>foreignObject>section .task-list-item.enabled label{cursor:pointer}div#\:\$p>svg>foreignObject>section .task-list-item+.task-list-item{margin-top:4px}div#\:\$p>svg>foreignObject>section .task-list-item .handle{display:none}div#\:\$p>svg>foreignObject>section .task-list-item-checkbox{margin:0 .2em .25em -1.4em;vertical-align:middle}div#\:\$p>svg>foreignObject>section .contains-task-list:dir(rtl) .task-list-item-checkbox{margin:0 -1.6em .25em .2em}div#\:\$p>svg>foreignObject>section .contains-task-list{position:relative}div#\:\$p>svg>foreignObject>section .contains-task-list:focus-within .task-list-item-convert-container,div#\:\$p>svg>foreignObject>section .contains-task-list:hover .task-list-item-convert-container{clip:auto;display:block;height:24px;overflow:visible;width:auto}div#\:\$p>svg>foreignObject>section ::-webkit-calendar-picker-indicator{filter:invert(50%)}div#\:\$p>svg>foreignObject>section :is(h1,marp-h1){color:var(--h1-color);font-size:1.6em}div#\:\$p>svg>foreignObject>section :is(h1,marp-h1),div#\:\$p>svg>foreignObject>section :is(h2,marp-h2){border-bottom:none}div#\:\$p>svg>foreignObject>section :is(h2,marp-h2){font-size:1.3em}div#\:\$p>svg>foreignObject>section :is(h3,marp-h3){font-size:1.1em}div#\:\$p>svg>foreignObject>section :is(h4,marp-h4){font-size:1.05em}div#\:\$p>svg>foreignObject>section :is(h5,marp-h5){font-size:1em}div#\:\$p>svg>foreignObject>section :is(h6,marp-h6){font-size:.9em}div#\:\$p>svg>foreignObject>section :is(h1,marp-h1) strong,div#\:\$p>svg>foreignObject>section :is(h2,marp-h2) strong,div#\:\$p>svg>foreignObject>section :is(h3,marp-h3) strong,div#\:\$p>svg>foreignObject>section :is(h4,marp-h4) strong,div#\:\$p>svg>foreignObject>section :is(h5,marp-h5) strong,div#\:\$p>svg>foreignObject>section :is(h6,marp-h6) strong{color:var(--heading-strong-color);font-weight:inherit}div#\:\$p>svg>foreignObject>section :is(h1,marp-h1)::part(auto-scaling),div#\:\$p>svg>foreignObject>section :is(h2,marp-h2)::part(auto-scaling),div#\:\$p>svg>foreignObject>section :is(h3,marp-h3)::part(auto-scaling),div#\:\$p>svg>foreignObject>section :is(h4,marp-h4)::part(auto-scaling),div#\:\$p>svg>foreignObject>section :is(h5,marp-h5)::part(auto-scaling),div#\:\$p>svg>foreignObject>section :is(h6,marp-h6)::part(auto-scaling){max-height:563px}div#\:\$p>svg>foreignObject>section hr{height:0;padding-top:.25em}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre){border:1px solid var(--color-border-default);line-height:1.15;overflow:visible}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre)::part(auto-scaling){max-height:529px}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs){color:var(--color-prettylights-syntax-storage-modifier-import)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-doctag),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-keyword),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-meta .hljs-keyword),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-template-tag),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-template-variable),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-type),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-variable.language_){color:var(--color-prettylights-syntax-keyword)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-title),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-title.class_),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-title.class_.inherited__),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-title.function_){color:var(--color-prettylights-syntax-entity)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-attr),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-attribute),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-literal),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-meta),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-number),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-operator),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-selector-attr),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-selector-class),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-selector-id),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-variable){color:var(--color-prettylights-syntax-constant)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-meta .hljs-string),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-regexp),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-string){color:var(--color-prettylights-syntax-string)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-built_in),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-symbol){color:var(--color-prettylights-syntax-variable)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-code),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-comment),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-formula){color:var(--color-prettylights-syntax-comment)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-name),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-quote),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-selector-pseudo),div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-selector-tag){color:var(--color-prettylights-syntax-entity-tag)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-subst){color:var(--color-prettylights-syntax-storage-modifier-import)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-section){color:var(--color-prettylights-syntax-markup-heading);font-weight:700}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-bullet){color:var(--color-prettylights-syntax-markup-list)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-emphasis){color:var(--color-prettylights-syntax-markup-italic);font-style:italic}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-strong){color:var(--color-prettylights-syntax-markup-bold);font-weight:700}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-addition){background-color:var(--color-prettylights-syntax-markup-inserted-bg);color:var(--color-prettylights-syntax-markup-inserted-text)}div#\:\$p>svg>foreignObject>section :is(pre,marp-pre) :where(.hljs-deletion){background-color:var(--color-prettylights-syntax-markup-deleted-bg);color:var(--color-prettylights-syntax-markup-deleted-text)}div#\:\$p>svg>foreignObject>section footer,div#\:\$p>svg>foreignObject>section header{color:var(--header-footer-color);font-size:18px;left:30px;margin:0;position:absolute}div#\:\$p>svg>foreignObject>section header{top:21px}div#\:\$p>svg>foreignObject>section footer{bottom:21px}div#\:\$p>svg>foreignObject>section{--h1-color:#246;--header-footer-color:hsla(0,0%,40%,.75);--heading-strong-color:#48c;--paginate-color:#777;align-items:stretch;display:flex;flex-flow:column nowrap;font-size:29px;height:720px;justify-content:center;padding:78.5px;width:1280px}div#\:\$p>svg>foreignObject>section{--marpit-root-font-size:29px}div#\:\$p>svg>foreignObject>section:where(.invert){--h1-color:#cee7ff;--header-footer-color:hsla(0,0%,60%,.75);--heading-strong-color:#7bf;--paginate-color:#999}div#\:\$p>svg>foreignObject>section>:last-child,div#\:\$p>svg>foreignObject>section[data-footer]>:nth-last-child(2){margin-bottom:0}div#\:\$p>svg>foreignObject>section>:first-child,div#\:\$p>svg>foreignObject>section>header:first-child+*{margin-top:0}div#\:\$p>svg>foreignObject>section:after{bottom:21px;color:var(--paginate-color);font-size:24px;padding:0;position:absolute;right:30px}div#\:\$p>svg>foreignObject>section:after{--marpit-root-font-size:24px}div#\:\$p>svg>foreignObject>section[data-color] :is(h1,marp-h1),div#\:\$p>svg>foreignObject>section[data-color] :is(h2,marp-h2),div#\:\$p>svg>foreignObject>section[data-color] :is(h3,marp-h3),div#\:\$p>svg>foreignObject>section[data-color] :is(h4,marp-h4),div#\:\$p>svg>foreignObject>section[data-color] :is(h5,marp-h5),div#\:\$p>svg>foreignObject>section[data-color] :is(h6,marp-h6){color:currentcolor}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background]{columns:initial!important;display:block!important;padding:0!important}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background]:after,div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background]:before,div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=content]:after,div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=content]:before{display:none!important}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background]>div[data-marpit-advanced-background-container]{all:initial;display:flex;flex-direction:row;height:100%;overflow:hidden;width:100%}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background]>div[data-marpit-advanced-background-container][data-marpit-advanced-background-direction=vertical]{flex-direction:column}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background][data-marpit-advanced-background-split]>div[data-marpit-advanced-background-container]{width:var(--marpit-advanced-background-split,50%)}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background][data-marpit-advanced-background-split=right]>div[data-marpit-advanced-background-container]{margin-left:calc(100% - var(--marpit-advanced-background-split, 50%))}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background]>div[data-marpit-advanced-background-container]>figure{all:initial;background-position:center;background-repeat:no-repeat;background-size:cover;flex:auto;margin:0}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=background]>div[data-marpit-advanced-background-container]>figure>figcaption{position:absolute;border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;white-space:nowrap;width:1px}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=content],div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=pseudo]{background:transparent!important}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background=pseudo],div#\:\$p>svg[data-marpit-svg]>foreignObject[data-marpit-advanced-background=pseudo]{pointer-events:none!important}div#\:\$p>svg>foreignObject>section[data-marpit-advanced-background-split]{width:100%;height:100%}</style></head><body><div class="bespoke-marp-osc"><button data-bespoke-marp-osc="prev" tabindex="-1" title="Previous slide">Previous slide</button><span data-bespoke-marp-osc="page"></span><button data-bespoke-marp-osc="next" tabindex="-1" title="Next slide">Next slide</button><button data-bespoke-marp-osc="fullscreen" tabindex="-1" title="Toggle fullscreen (f)">Toggle fullscreen</button><button data-bespoke-marp-osc="presenter" tabindex="-1" title="Open presenter view (p)">Open presenter view</button></div><div id=":$p"><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="1" data-paginate="true" data-class="invert" data-heading-divider="3" lang="en-US" class="invert" data-marpit-pagination="1" style="--paginate:true;--class:invert;--heading-divider:3;" data-marpit-pagination-total="25">
<h1 id="1-vectors">1. Vectors</h1>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="2" data-marpit-fragments="7" data-paginate="true" data-class="invert" data-heading-divider="3" lang="en-US" class="invert" data-marpit-pagination="2" style="--paginate:true;--class:invert;--heading-divider:3;" data-marpit-pagination-total="25">
<h2 id="what-is-a-vector">What is a vector?</h2>
<ul>
<li data-marpit-fragment="1">Vector is a mathematical thingy with a <strong>length</strong> and a <strong>direction</strong></li>
<li data-marpit-fragment="2">Often represented by an arrow (see next slide)</li>
<li data-marpit-fragment="3">Can have multiple dimensions (also called components)
<ul>
<li data-marpit-fragment="4">In our case usually 2 or 3:</li>
<li data-marpit-fragment="5"><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></g></g></g></svg></mjx-container>, <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></g></g></g></svg></mjx-container>, and possibly <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.052ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 465 453"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D467" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path></g></g></g></svg></mjx-container></li>
</ul>
</li>
<li data-marpit-fragment="6">Video games are full of vectors!
<ul>
<li data-marpit-fragment="7">Used for depicting position, velocity, acceleration, forces....</li>
</ul>
</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section data-marpit-fragments="5" data-paginate="true" data-class="invert" data-heading-divider="3" lang="en-US" class="invert" data-marpit-pagination="3" style="--paginate:true;--class:invert;--heading-divider:3;--marpit-advanced-background-split:50%;" data-marpit-pagination-total="25" data-marpit-advanced-background="background" data-marpit-advanced-background-split="right"><div data-marpit-advanced-background-container="true" data-marpit-advanced-background-direction="horizontal"><figure style="background-image:url("imgs/2d-vector.png");background-size:90%;"><figcaption>width:</figcaption></figure></div></section></foreignObject><foreignObject width="50%" height="720"><section id="3" data-marpit-fragments="5" data-paginate="true" data-class="invert" data-heading-divider="3" lang="en-US" class="invert" data-marpit-pagination="3" style="--paginate:true;--class:invert;--heading-divider:3;--marpit-advanced-background-split:50%;" data-marpit-pagination-total="25" data-marpit-advanced-background="content" data-marpit-advanced-background-split="right">
<h3 id="2d-vector-example">2D vector example</h3>
<ul>
<li data-marpit-fragment="1">
<p>This is a 2D vector <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="9.743ex" height="3.097ex" role="img" focusable="false" viewBox="0 -1119 4306.2 1369"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></g><g data-mml-node="mo" transform="translate(514,305) translate(-250 0)"><path data-c="20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 653 460 633Q471 626 471 615Q471 606 468 603T454 594Q411 572 379 531Q377 529 374 525T369 519T364 517T357 516Q350 516 344 521T337 536Q337 555 384 595H213L42 596Q29 605 29 615Q29 622 42 635H401Q377 673 377 694Z"></path></g></g></g><g data-mml-node="mo" transform="translate(1027.8,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path></g><g data-mml-node="mo" transform="translate(2083.6,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path></g><g data-mml-node="mn" transform="translate(2472.6,0)"><path data-c="36" d="M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601Q351 623 311 626Q240 626 194 566Q147 500 147 364L148 360Q153 366 156 373Q197 433 263 433H267Q313 433 348 414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 34T253 -22Q225 -22 199 -14T143 16T92 75T56 172T42 313ZM257 397Q227 397 205 380T171 335T154 278T148 216Q148 133 160 97T198 39Q222 21 251 21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z"></path></g><g data-mml-node="mo" transform="translate(2972.6,0)"><path data-c="2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path></g><g data-mml-node="mn" transform="translate(3417.2,0)"><path data-c="33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></g><g data-mml-node="mo" transform="translate(3917.2,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></g></g></g></svg></mjx-container></p>
<ul>
<li data-marpit-fragment="2">x-component <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.357ex;" xmlns="http://www.w3.org/2000/svg" width="6.948ex" height="2.889ex" role="img" focusable="false" viewBox="0 -1119 3071 1276.8"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></g><g data-mml-node="mo" transform="translate(514,305) translate(-250 0)"><path data-c="20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 653 460 633Q471 626 471 615Q471 606 468 603T454 594Q411 572 379 531Q377 529 374 525T369 519T364 517T357 516Q350 516 344 521T337 536Q337 555 384 595H213L42 596Q29 605 29 615Q29 622 42 635H401Q377 673 377 694Z"></path></g></g></g><g data-mml-node="mi" transform="translate(783,-150) scale(0.707)"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></g></g><g data-mml-node="mo" transform="translate(1515.2,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path></g><g data-mml-node="mn" transform="translate(2571,0)"><path data-c="36" d="M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601Q351 623 311 626Q240 626 194 566Q147 500 147 364L148 360Q153 366 156 373Q197 433 263 433H267Q313 433 348 414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 34T253 -22Q225 -22 199 -14T143 16T92 75T56 172T42 313ZM257 397Q227 397 205 380T171 335T154 278T148 216Q148 133 160 97T198 39Q222 21 251 21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z"></path></g></g></g></svg></mjx-container></li>
<li data-marpit-fragment="3">y-component <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.667ex;" xmlns="http://www.w3.org/2000/svg" width="6.817ex" height="3.199ex" role="img" focusable="false" viewBox="0 -1119 3013 1414"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></g><g data-mml-node="mo" transform="translate(514,305) translate(-250 0)"><path data-c="20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 653 460 633Q471 626 471 615Q471 606 468 603T454 594Q411 572 379 531Q377 529 374 525T369 519T364 517T357 516Q350 516 344 521T337 536Q337 555 384 595H213L42 596Q29 605 29 615Q29 622 42 635H401Q377 673 377 694Z"></path></g></g></g><g data-mml-node="mi" transform="translate(783,-150) scale(0.707)"><path data-c="1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></g></g><g data-mml-node="mo" transform="translate(1457.3,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path></g><g data-mml-node="mn" transform="translate(2513,0)"><path data-c="33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path></g></g></g></svg></mjx-container></li>
</ul>
</li>
<li data-marpit-fragment="4">
<p>Vectors start from the origin, or <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="5.029ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2222.7 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path></g><g data-mml-node="mn" transform="translate(389,0)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></g><g data-mml-node="mo" transform="translate(889,0)"><path data-c="2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path></g><g data-mml-node="mn" transform="translate(1333.7,0)"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></g><g data-mml-node="mo" transform="translate(1833.7,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></g></g></g></svg></mjx-container></p>
</li>
<li data-marpit-fragment="5">
<p>In C#:</p>
<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c#">Vector2 vectorA = <span class="hljs-keyword">new</span> Vector2(<span class="hljs-number">6</span>,<span class="hljs-number">3</span>);
</code></pre>
</li>
</ul>
</section>
</foreignObject><foreignObject width="1280" height="720" data-marpit-advanced-background="pseudo"><section data-marpit-fragments="5" data-paginate="true" data-class="invert" data-heading-divider="3" lang="en-US" class="invert" data-marpit-pagination="3" style="" data-marpit-pagination-total="25" data-marpit-advanced-background="pseudo" data-marpit-advanced-background-split="right"></section></foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="4" data-marpit-fragments="5" data-paginate="true" data-class="invert" data-heading-divider="3" lang="en-US" class="invert" data-marpit-pagination="4" style="--paginate:true;--class:invert;--heading-divider:3;" data-marpit-pagination-total="25">
<h3 id="2d-vector-length">2D vector length</h3>
<ul>
<li data-marpit-fragment="1">
<p>The length of a 2D vector is given by the Pythagoras theorem</p>
<ul>
<li data-marpit-fragment="2"><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -1.061ex;" xmlns="http://www.w3.org/2000/svg" width="16.529ex" height="4.208ex" role="img" focusable="false" viewBox="0 -1391 7306 1860"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo" transform="translate(0 -0.5)"><path data-c="7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(278,0)"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></g><g data-mml-node="mo" transform="translate(514,305) translate(-250 0)"><path data-c="20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 653 460 633Q471 626 471 615Q471 606 468 603T454 594Q411 572 379 531Q377 529 374 525T369 519T364 517T357 516Q350 516 344 521T337 536Q337 555 384 595H213L42 596Q29 605 29 615Q29 622 42 635H401Q377 673 377 694Z"></path></g></g></g><g data-mml-node="mo" transform="translate(1028,0) translate(0 -0.5)"><path data-c="7C" d="M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z"></path></g><g data-mml-node="mo" transform="translate(1583.8,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 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</ul>
</li>
<li data-marpit-fragment="4">
<p>In C#:</p>
<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c#"><span class="hljs-built_in">float</span> length = Mathf.Sqrt(Mathf.Pow(vectorA.x, <span class="hljs-number">2</span>) + Mathf.Pow(vectorA.y, <span class="hljs-number">2</span>));
</code></pre>
<ul>
<li data-marpit-fragment="5">Vector classes have a shorthand, too: <code>A.magnitude</code></li>
</ul>
</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="5" data-marpit-fragments="9" data-paginate="true" data-class="invert" data-heading-divider="3" lang="en-US" class="invert" data-marpit-pagination="5" style="--paginate:true;--class:invert;--heading-divider:3;" data-marpit-pagination-total="25">
<h2 id="vector-arithmetic">Vector arithmetic</h2>
<ul>
<li data-marpit-fragment="1"><a href="https://docs.unity3d.com/2019.3/Documentation/Manual/UnderstandingVectorArithmetic.html">Manual: Understanding Vector Arithmetic</a></li>
<li data-marpit-fragment="2">Let's introduce the most important vector operations
<ul>
<li data-marpit-fragment="3">Addition</li>
<li data-marpit-fragment="4">Subtraction</li>
<li data-marpit-fragment="5">Scalar multiplication</li>
<li data-marpit-fragment="6">Extra: vector multiplication
<ul>
<li data-marpit-fragment="7">Dot product</li>
<li data-marpit-fragment="8">Cross product (WIP)</li>
</ul>
</li>
</ul>
</li>
<li data-marpit-fragment="9">C# examples included</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section data-marpit-fragments="3" data-paginate="true" data-class="invert" data-heading-divider="3" lang="en-US" class="invert" data-marpit-pagination="6" style="--paginate:true;--class:invert;--heading-divider:3;--marpit-advanced-background-split:50%;" data-marpit-pagination-total="25" data-marpit-advanced-background="background" data-marpit-advanced-background-split="right"><div data-marpit-advanced-background-container="true" data-marpit-advanced-background-direction="horizontal"><figure style="background-image:url("imgs/2d-vector-addition.png");background-size:90%;"><figcaption>width:</figcaption></figure></div></section></foreignObject><foreignObject width="50%" height="720"><section id="6" data-marpit-fragments="3" data-paginate="true" data-class="invert" data-heading-divider="3" lang="en-US" class="invert" data-marpit-pagination="6" style="--paginate:true;--class:invert;--heading-divider:3;--marpit-advanced-background-split:50%;" data-marpit-pagination-total="25" data-marpit-advanced-background="content" data-marpit-advanced-background-split="right">
<h3 id="vector-addition">Vector addition</h3>
<ul>
<li data-marpit-fragment="1">
<p>sum of two vectors is calculated by summing up the individual components</p>
</li>
<li data-marpit-fragment="2">
<p><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="10.916ex" height="2.717ex" role="img" focusable="false" viewBox="0 -1119 4825 1201"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D436" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"></path></g><g data-mml-node="mo" transform="translate(474.5,294) translate(-250 0)"><path data-c="20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 653 460 633Q471 626 471 615Q471 606 468 603T454 594Q411 572 379 531Q377 529 374 525T369 519T364 517T357 516Q350 516 344 521T337 536Q337 555 384 595H213L42 596Q29 605 29 615Q29 622 42 635H401Q377 673 377 694Z"></path></g></g></g><g data-mml-node="mo" transform="translate(1037.8,0)"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(2093.6,0)"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></g><g data-mml-node="mo" transform="translate(514,305) translate(-250 0)"><path data-c="20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 653 460 633Q471 626 471 615Q471 606 468 603T454 594Q411 572 379 531Q377 529 374 525T369 519T364 517T357 516Q350 516 344 521T337 536Q337 555 384 595H213L42 596Q29 605 29 615Q29 622 42 635H401Q377 673 377 694Z"></path></g></g></g><g data-mml-node="mo" transform="translate(3065.8,0)"><path data-c="2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(4066,0)"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path></g><g data-mml-node="mo" transform="translate(462.8,272) translate(-250 0)"><path data-c="20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 653 460 633Q471 626 471 615Q471 606 468 603T454 594Q411 572 379 531Q377 529 374 525T369 519T364 517T357 516Q350 516 344 521T337 536Q337 555 384 595H213L42 596Q29 605 29 615Q29 622 42 635H401Q377 673 377 694Z"></path></g></g></g></g></g></svg></mjx-container><br />
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308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></g><g data-mml-node="mo" transform="translate(7112.6,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></g></g></g></svg></mjx-container><br />
<mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="23.257ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 10279.6 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><path data-c="3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path></g><g data-mml-node="mo" transform="translate(1055.8,0)"><path data-c="28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path></g><g data-mml-node="mn" transform="translate(1444.8,0)"><path data-c="33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 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495Q356 495 341 509T326 548Q326 592 373 601Q351 623 311 626Q240 626 194 566Q147 500 147 364L148 360Q153 366 156 373Q197 433 263 433H267Q313 433 348 414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 34T253 -22Q225 -22 199 -14T143 16T92 75T56 172T42 313ZM257 397Q227 397 205 380T171 335T154 278T148 216Q148 133 160 97T198 39Q222 21 251 21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z"></path></g><g data-mml-node="mo" transform="translate(3667.2,0)"><path data-c="2C" d="M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z"></path></g><g data-mml-node="mn" transform="translate(4111.9,0)"><path data-c="33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 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6T161 3T152 1T140 0Q113 0 96 17Z"></path></g><g data-mml-node="mn" transform="translate(9390.6,0)"><path data-c="31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></g><g data-mml-node="mo" transform="translate(9890.6,0)"><path data-c="29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></g></g></g></svg></mjx-container></p>
</li>
<li data-marpit-fragment="3">
<p>can be illustrated by moving <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.717ex" height="2.457ex" role="img" focusable="false" viewBox="0 -1086 759 1086"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path></g><g data-mml-node="mo" transform="translate(462.8,272) translate(-250 0)"><path data-c="20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 653 460 633Q471 626 471 615Q471 606 468 603T454 594Q411 572 379 531Q377 529 374 525T369 519T364 517T357 516Q350 516 344 521T337 536Q337 555 384 595H213L42 596Q29 605 29 615Q29 622 42 635H401Q377 673 377 694Z"></path></g></g></g></g></g></svg></mjx-container> to start from the endpoint of <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="2.532ex" role="img" focusable="false" viewBox="0 -1119 750 1119"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></g><g data-mml-node="mo" transform="translate(514,305) translate(-250 0)"><path data-c="20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 653 460 633Q471 626 471 615Q471 606 468 603T454 594Q411 572 379 531Q377 529 374 525T369 519T364 517T357 516Q350 516 344 521T337 536Q337 555 384 595H213L42 596Q29 605 29 615Q29 622 42 635H401Q377 673 377 694Z"></path></g></g></g></g></g></svg></mjx-container></p>
<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c#">Vector2 A = <span class="hljs-keyword">new</span> Vector2(<span class="hljs-number">3.0f</span>, <span class="hljs-number">3.0f</span>);
Vector2 B = <span class="hljs-keyword">new</span> Vector2(<span class="hljs-number">6.0f</span>,<span class="hljs-number">-2.0f</span>);
Vector2 C = A + B;
</code></pre>
</li>
</ul>
</section>
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<h3 id="vector-subtraction">Vector subtraction</h3>
<ul>
<li data-marpit-fragment="1">
<p>difference of two vectors, <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="6.18ex" height="2.717ex" role="img" focusable="false" viewBox="0 -1119 2731.4 1201"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></path></g><g data-mml-node="mo" transform="translate(514,305) translate(-250 0)"><path data-c="20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 653 460 633Q471 626 471 615Q471 606 468 603T454 594Q411 572 379 531Q377 529 374 525T369 519T364 517T357 516Q350 516 344 521T337 536Q337 555 384 595H213L42 596Q29 605 29 615Q29 622 42 635H401Q377 673 377 694Z"></path></g></g></g><g data-mml-node="mo" transform="translate(972.2,0)"><path data-c="2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(1972.4,0)"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path></g><g data-mml-node="mo" transform="translate(462.8,272) translate(-250 0)"><path data-c="20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 653 460 633Q471 626 471 615Q471 606 468 603T454 594Q411 572 379 531Q377 529 374 525T369 519T364 517T357 516Q350 516 344 521T337 536Q337 555 384 595H213L42 596Q29 605 29 615Q29 622 42 635H401Q377 673 377 694Z"></path></g></g></g></g></g></svg></mjx-container></p>
<ul>
<li data-marpit-fragment="2"><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="3.477ex" height="2.643ex" role="img" focusable="false" viewBox="0 -1086 1537 1168"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><path data-c="2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(778,0)"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path></g><g data-mml-node="mo" transform="translate(462.8,272) translate(-250 0)"><path data-c="20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 653 460 633Q471 626 471 615Q471 606 468 603T454 594Q411 572 379 531Q377 529 374 525T369 519T364 517T357 516Q350 516 344 521T337 536Q337 555 384 595H213L42 596Q29 605 29 615Q29 622 42 635H401Q377 673 377 694Z"></path></g></g></g></g></g></svg></mjx-container> is is "flipped" <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.717ex" height="2.457ex" role="img" focusable="false" viewBox="0 -1086 759 1086"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path></g><g data-mml-node="mo" transform="translate(462.8,272) translate(-250 0)"><path data-c="20D7" d="M377 694Q377 702 382 708T397 714Q404 714 409 709Q414 705 419 690Q429 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</ul>
</li>
<li data-marpit-fragment="4">
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<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c#">Vector2 A = <span class="hljs-keyword">new</span> Vector2(<span class="hljs-number">3.0f</span>, <span class="hljs-number">3.0f</span>);
Vector2 B = <span class="hljs-keyword">new</span> Vector2(<span class="hljs-number">6.0f</span>,<span class="hljs-number">-2.0f</span>);
Vector2 C = A - B;
</code></pre>
</li>
</ul>
</section>
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<h3 id="scalar-multiplication">Scalar multiplication</h3>
<ul>
<li data-marpit-fragment="1">
<p>When a vector is multiplied by a scalar (a number), the vector is <em>scaled</em></p>
<ul>
<li data-marpit-fragment="2">If a vector is multiplied by 2, its length doubles<br />
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</ul>
<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c#">Vector2 A = <span class="hljs-keyword">new</span> Vector2(<span class="hljs-number">3.0f</span>,<span class="hljs-number">3.0f</span>);
Vector2 C = <span class="hljs-number">2</span> * A;
</code></pre>
</li>
</ul>
</section>
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<h3 id="special-cases-for-scalar-multiplication">Special cases for scalar multiplication</h3>
<ul>
<li data-marpit-fragment="1">
<p>If the scalar is negative, the vector gets flipped</p>
<ul>
<li data-marpit-fragment="2">that's what happened in subtraction</li>
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</ul>
</li>
<li data-marpit-fragment="4">
<p>What about division?</p>
<ul>
<li data-marpit-fragment="5">it's basically multiplication as well</li>
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</ul>
<p><img src="imgs/2d-vector-scalar-multiplication-2.png" alt="" /></p>
</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="10" data-paginate="true" data-class="invert" data-heading-divider="3" lang="en-US" class="invert" data-marpit-pagination="10" style="--paginate:true;--class:invert;--heading-divider:3;" data-marpit-pagination-total="25">
<h1 id="vectors-in-unity">Vectors in Unity</h1>
</section>
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<h2 id="vectors-in-video-games">Vectors in video games</h2>
<ul>
<li data-marpit-fragment="1">position</li>
<li data-marpit-fragment="2">velocity</li>
<li data-marpit-fragment="3">acceleration</li>
<li data-marpit-fragment="4">rotation</li>
<li data-marpit-fragment="5">forces</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="12" data-marpit-fragments="10" data-paginate="true" data-class="invert" data-heading-divider="3" lang="en-US" class="invert" data-marpit-pagination="12" style="--paginate:true;--class:invert;--heading-divider:3;" data-marpit-pagination-total="25">
<h2 id="vector-classes">Vector classes</h2>
<ul>
<li data-marpit-fragment="1">
<p>Unity has two Vector classes, <a href="https://docs.unity3d.com/ScriptReference/Vector2.html">Vector2</a> and <a href="https://docs.unity3d.com/ScriptReference/Vector3.html">Vector3</a></p>
<ul>
<li data-marpit-fragment="2">"2" and "3" here are the number of dimensions: (x,y) and (x,y,z)</li>
<li data-marpit-fragment="3">
<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c#">Vector2 position = <span class="hljs-keyword">new</span> Vector2(<span class="hljs-number">1.0f</span>, <span class="hljs-number">2.0f</span>)
</code></pre>
</li>
</ul>
</li>
<li data-marpit-fragment="4">
<p>Vector components can be accessed with the dot notation:</p>
<ul>
<li data-marpit-fragment="5"><code>position.x</code>, <code>position.y</code>, <code>position.z</code></li>
</ul>
</li>
<li data-marpit-fragment="6">
<p>Vectors can't be directly modified:</p>
<ul>
<li data-marpit-fragment="7">No: <code>position.x = 3.0f</code></li>
<li data-marpit-fragment="8">
<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c#">position = <span class="hljs-keyword">new</span> Vector2(<span class="hljs-number">3.0f</span>, position.y);
</code></pre>
</li>
</ul>
</li>
<li data-marpit-fragment="9">
<p>Length of a vector can be acquired with <code>vector.magnitude</code></p>
<ul>
<li data-marpit-fragment="10"><a href="https://docs.unity3d.com/ScriptReference/Vector3-magnitude.html">Script Reference: Vector3 magnitude</a></li>
</ul>
</li>
</ul>
</section>
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<h2 id="velocity-vector">Velocity vector</h2>
<ul>
<li data-marpit-fragment="1">Velocity is an important concept in game development!</li>
<li data-marpit-fragment="2">It's the <em><strong>rate of change</strong></em> of position
<ul>
<li data-marpit-fragment="3"><code>new_position = old_position + velocity;</code></li>
</ul>
</li>
<li data-marpit-fragment="4">3-dimensional velocity vector can be added like this to GameObject's position in an <code>Update()</code> method:
<ul>
<li data-marpit-fragment="5"><code>transform.position += velocity;</code></li>
<li data-marpit-fragment="6">We can use player input, collision, etc. to adjust the velocity as needed</li>
</ul>
</li>
</ul>
</section>
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<p><em><strong>Note:</strong></em> Velocity vector is usually drawn so it starts from the moving object</p>
<ul>
<li data-marpit-fragment="1">
<p>(Remember, though, that vectors do not "know" its starting positions.)</p>
<p><img src="imgs/velocity-vector.png" alt="" /></p>
</li>
</ul>
</section>
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<h2 id="acceleration-vector">Acceleration vector</h2>
<ul>
<li data-marpit-fragment="1">Another important concept is <em><strong>acceleration</strong></em></li>
<li data-marpit-fragment="2">It's the rate of change of a GameObject's velocity!
<ul>
<li data-marpit-fragment="3"><code>new_velocity = old_velocity + acceleration</code>;</li>
</ul>
</li>
<li data-marpit-fragment="4">We can use acceleration to change Rigidbody's velocity
<ul>
<li data-marpit-fragment="5"><code>rb.velocity += acceleration</code>;</li>
</ul>
</li>
<li data-marpit-fragment="6">We can't use that for GameObjects without rigidbodies, but we can store velocity in a separate variable<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c#">velocity += acceleration;
transform.position += velocity;
</code></pre>
</li>
<li data-marpit-fragment="7">This might seem like a cheap trick, but actually allows us to create <em><strong>smoother motion</strong></em> if we use player input, collision, etc to control acceleration instead of velocity</li>
</ul>
</section>
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<h2 id="distance-vector">Distance vector</h2>
<ul>
<li data-marpit-fragment="1">To get the distance vector between two objects, we use vector subtraction
<ul>
<li data-marpit-fragment="2"><code>vector_B - vector_A</code></li>
<li data-marpit-fragment="3"><code>transform.position - otherGameObject.transform.position;</code></li>
</ul>
</li>
<li data-marpit-fragment="4">To just get the length of the distance vector, a.k.a, the distance:
<ul>
<li data-marpit-fragment="5"><code>Vector2.Distance(vector_A, vector_B)</code></li>
</ul>
</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="17" data-marpit-fragments="4" data-paginate="true" data-class="invert" data-heading-divider="3" lang="en-US" class="invert" data-marpit-pagination="17" style="--paginate:true;--class:invert;--heading-divider:3;" data-marpit-pagination-total="25">
<h2 id="special-vectors-of-unity">Special vectors of Unity</h2>
<ul>
<li data-marpit-fragment="1">See Static properties in <a href="https://docs.unity3d.com/ScriptReference/Vector3.html">Script Reference: Vector3</a>
<ul>
<li data-marpit-fragment="2"><code>Vector3.right</code>: the global <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></g></g></g></svg></mjx-container> axis</li>
<li data-marpit-fragment="3"><code>Vector3.up</code>: the global <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.464ex;" xmlns="http://www.w3.org/2000/svg" width="1.109ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 490 647"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></g></g></g></svg></mjx-container> axis</li>
<li data-marpit-fragment="4"><code>Vector3.forward</code>: the global <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.025ex;" xmlns="http://www.w3.org/2000/svg" width="1.052ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 465 453"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><path data-c="1D467" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path></g></g></g></svg></mjx-container> axis</li>
</ul>
</li>
</ul>
</section>
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<h2 id="common-vector-operations">Common vector operations</h2>
</section>
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<h3 id="normalizing-a-vector">Normalizing a vector</h3>
<ul>
<li data-marpit-fragment="1">When we are not interested about the length of a vector, only its <strong>direction</strong>, it helps to <em>normalize</em> the vector</li>
<li data-marpit-fragment="2">Normalizing means setting the length of a vector to be <mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><path data-c="31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></g></g></g></svg></mjx-container></li>
<li data-marpit-fragment="3">This is achieved by dividing the vector by its length</li>
<li data-marpit-fragment="4"><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.566ex;" xmlns="http://www.w3.org/2000/svg" width="18.743ex" height="3.097ex" role="img" focusable="false" viewBox="0 -1119 8284.3 1369"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><path data-c="1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 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<li data-marpit-fragment="5">in C#, the normalized version of any vector can be easily accessed:<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c#">Vector2 UnitVector = A.normalized;
</code></pre>
</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="20" data-marpit-fragments="5" data-paginate="true" data-class="invert" data-heading-divider="3" lang="en-US" class="invert" data-marpit-pagination="20" style="--paginate:true;--class:invert;--heading-divider:3;" data-marpit-pagination-total="25">
<h3 id="rotating-a-vector">Rotating a vector</h3>
<ul>
<li data-marpit-fragment="1">In Unity, rotation is represented by <a href="5-math-classes#quaternion">Quaternions</a> (fin. kvaternio, kvaterniot)</li>
<li data-marpit-fragment="2">To rotate a vector by a given angle, you can do a <em><strong>Quaternion rotation operation</strong></em>:<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c#">Vector3 rotatedVector = Quaternion.Euler(<span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">90</span>) * originalVector;
</code></pre>
<ul>
<li data-marpit-fragment="3">This isn't a regular multiplication, so do note that Quaternion.Euler has to be on the <em><strong>left side</strong></em> of the vector.</li>
</ul>
</li>
<li data-marpit-fragment="4">To rotate a Quaternion:<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c#">myQuaternion *= Quaternion.Euler(<span class="hljs-number">0</span>, <span class="hljs-number">0</span>, <span class="hljs-number">90</span>);
</code></pre>
</li>
<li data-marpit-fragment="5"><em><strong>Note:</strong></em> the Transform and Quaternion classes have many rotation methods available, see <a href="../unity-cookbook/transform#rotation">Transform Class: Rotation</a></li>
</ul>
</section>
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<h2 id="extra-dot-product---vectors-alignment-with-another-vector">Extra: Dot product - Vector's alignment with another vector</h2>
<ul>
<li data-marpit-fragment="1">If you want to know how much two vectors point to the same direction, we can use the <em><strong>dot product</strong></em>
<ul>
<li data-marpit-fragment="2"><a href="https://docs.unity3d.com/ScriptReference/Vector3.Dot.html">Script Reference: Vector3.Dot</a></li>
</ul>
</li>
<li data-marpit-fragment="3">The dot product returns a number, not a vector!
<ul>
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</ul>
</li>
<li data-marpit-fragment="5">For normalized vectors, <code>Vector3.Dot</code> returns
<ul>
<li data-marpit-fragment="6"><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: 0;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.507ex" role="img" focusable="false" viewBox="0 -666 500 666"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><path data-c="31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></g></g></g></svg></mjx-container> if they point in exactly the same direction</li>
<li data-marpit-fragment="7"><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="2.891ex" height="1.692ex" role="img" focusable="false" viewBox="0 -666 1278 748"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><path data-c="2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path></g><g data-mml-node="mn" transform="translate(778,0)"><path data-c="31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></g></g></g></svg></mjx-container> if they point in completely opposite directions</li>
<li data-marpit-fragment="8"><mjx-container class="MathJax" jax="SVG"><svg style="vertical-align: -0.05ex;" xmlns="http://www.w3.org/2000/svg" width="1.131ex" height="1.557ex" role="img" focusable="false" viewBox="0 -666 500 688"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mn"><path data-c="30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path></g></g></g></svg></mjx-container> if the vectors are perpendicular</li>
</ul>
</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="22" data-marpit-fragments="1" data-paginate="true" data-class="invert" data-background-color="#5d275d" data-heading-divider="3" lang="en-US" class="invert" data-marpit-pagination="22" style="--paginate:true;--class:invert;--background-color:#5d275d;--heading-divider:3;background-color:#5d275d;background-image:none;" data-marpit-pagination-total="25">
<h3 id="dot-product-example">Dot product example</h3>
<ul>
<li data-marpit-fragment="1">Check if two rigidbodies are moving to the same direction:<pre is="marp-pre" data-auto-scaling="downscale-only"><code class="language-c#"><span class="hljs-built_in">float</span> alignment = Vector3.Dot(
_rigidBody.velocity.normalized,
otherGameObjectRigidBody.velocity.normalized)
</code></pre>
</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="23" data-marpit-fragments="2" data-paginate="true" data-class="invert" data-heading-divider="3" lang="en-US" class="invert" data-marpit-pagination="23" style="--paginate:true;--class:invert;--heading-divider:3;" data-marpit-pagination-total="25">
<h2 id="note-about-distance">Note about distance</h2>
<ul>
<li data-marpit-fragment="1">Note: when performance is important, using magnitude can be a bad idea: it includes the costly square root operation
<ul>
<li data-marpit-fragment="2">if you need to only compare magnitudes, or you're squaring it right away, use <code>.sqrMagnitude</code> instead!</li>
</ul>
</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="24" data-paginate="true" data-class="invert" data-background-color="#29366f" data-heading-divider="3" lang="en-US" class="invert" data-marpit-pagination="24" style="--paginate:true;--class:invert;--background-color:#29366f;--heading-divider:3;background-color:#29366f;background-image:none;" data-marpit-pagination-total="25">
<h2 id="exercise-1-im-being-avoided">Exercise 1. I'm being avoided</h2>
<p>Create a scene with two GameObjects, a player and a static enemy.<br />
Calculate <strong>the distance</strong> between the player and the enemy.</p>
<p>If the distance is smaller than some given threshold value,</p>
<ul>
<li><img class="emoji" draggable="false" alt="⭐" src="https://cdn.jsdelivr.net/gh/jdecked/[email protected]/assets/svg/2b50.svg" data-marp-twemoji=""/> Change the color of the enemy (and set the color back to default when you're no more on the range)</li>
<li><img class="emoji" draggable="false" alt="⭐" src="https://cdn.jsdelivr.net/gh/jdecked/[email protected]/assets/svg/2b50.svg" data-marp-twemoji=""/><img class="emoji" draggable="false" alt="⭐" src="https://cdn.jsdelivr.net/gh/jdecked/[email protected]/assets/svg/2b50.svg" data-marp-twemoji=""/> Make the enemy shoot at player!</li>
<li><img class="emoji" draggable="false" alt="⭐" src="https://cdn.jsdelivr.net/gh/jdecked/[email protected]/assets/svg/2b50.svg" data-marp-twemoji=""/><img class="emoji" draggable="false" alt="⭐" src="https://cdn.jsdelivr.net/gh/jdecked/[email protected]/assets/svg/2b50.svg" data-marp-twemoji=""/><img class="emoji" draggable="false" alt="⭐" src="https://cdn.jsdelivr.net/gh/jdecked/[email protected]/assets/svg/2b50.svg" data-marp-twemoji=""/> Move the enemy farther away from the player along the shortest possible path.</li>
</ul>
</section>
</foreignObject></svg><svg data-marpit-svg="" viewBox="0 0 1280 720"><foreignObject width="1280" height="720"><section id="25" data-marpit-fragments="3" data-paginate="true" data-class="invert" data-heading-divider="3" lang="en-US" class="invert" data-marpit-pagination="25" style="--paginate:true;--class:invert;--heading-divider:3;" data-marpit-pagination-total="25">
<h2 id="reading--watching">Reading & watching</h2>
<ul>
<li data-marpit-fragment="1">
<p><a href="https://docs.unity3d.com/2019.3/Documentation/Manual/VectorCookbook.html">Manual: Vector Cookbook</a></p>
</li>
<li data-marpit-fragment="2">
<p><a href="https://www.youtube.com/watch?v=wXI9_olSrqo">Brackeys video: Vectors</a></p>
</li>
<li data-marpit-fragment="3">
<p><a href="https://www.youtube.com/watch?v=MOYiVLEnhrw">Freya Holmér video: Vectors & Dot Product</a></p>
</li>
</ul>
</section>
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