e39465ad2f
new file: Files/flashplayer_32_sa.exe
new file: favicon.ico
new file: globe.gif
new file: imgs/download.png
new file: imgs/zuck.jpg
new file: index.html
new file: other.ico
new file: script.js
new file: site.webmanifest
new file: sitemap.html
new file: styles/backround.css
new file: styles/border.css
new file: styles/fonts/Titillium_Web/OFL.txt
new file: styles/fonts/Titillium_Web/TitilliumWeb-Black.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Bold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-BoldItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLight.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-ExtraLightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Italic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Light.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-LightItalic.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-Regular.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBold.ttf
new file: styles/fonts/Titillium_Web/TitilliumWeb-SemiBoldItalic.ttf
new file: styles/fonts/webfontkit-20221027-163353/generator_config.txt
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-163353/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/stylesheet.css
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-demo.html
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff
new file: styles/fonts/webfontkit-20221027-163353/titilliumweb-extralight-webfont.woff2
new file: styles/fonts/webfontkit-20221027-165950/generator_config.txt
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/grid_12-825-55-15.css
new file: styles/fonts/webfontkit-20221027-165950/specimen_files/specimen_stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/stylesheet.css
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-demo.html
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff
new file: styles/fonts/webfontkit-20221027-165950/titilliumweb-bold-webfont.woff2
new file: styles/style.css
new file: tools/2048/.gitignore
new file: tools/2048/.jshintrc
new file: tools/2048/CONTRIBUTING.md
new file: tools/2048/LICENSE.txt
new file: tools/2048/README.md
new file: tools/2048/Rakefile
new file: tools/2048/favicon.ico
new file: tools/2048/index.html
new file: tools/2048/js/animframe_polyfill.js
new file: tools/2048/js/application.js
new file: tools/2048/js/bind_polyfill.js
new file: tools/2048/js/classlist_polyfill.js
new file: tools/2048/js/game_manager.js
new file: tools/2048/js/grid.js
new file: tools/2048/js/html_actuator.js
new file: tools/2048/js/keyboard_input_manager.js
new file: tools/2048/js/local_storage_manager.js
new file: tools/2048/js/tile.js
new file: tools/2048/meta/apple-touch-icon.png
new file: tools/webretro/cores/neocd_libretro.js
new file: tools/webretro/cores/neocd_libretro.wasm
new file: tools/webretro/cores/nestopia_libretro.js
new file: tools/webretro/cores/nestopia_libretro.wasm
new file: tools/webretro/cores/o2em_libretro.js
new file: tools/webretro/cores/o2em_libretro.wasm
new file: tools/webretro/cores/opera_libretro.js
new file: tools/webretro/cores/opera_libretro.wasm
271 lines
11 KiB
HTML
271 lines
11 KiB
HTML
<!DOCTYPE html>
|
||
<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
|
||
|
||
<!-- Mirrored from www.mathsisfun.com/algebra/scalar-vector-matrix.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:49:35 GMT -->
|
||
<head>
|
||
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
|
||
|
||
<!-- #BeginEditable "doctitle" -->
|
||
<title>Scalar, Vector, Matrix</title>
|
||
<script language="JavaScript" type="text/javascript">reSpell = [["kilometers", "kilometres"]];</script>
|
||
<meta name="description" content="Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.">
|
||
<!-- #EndEditable -->
|
||
<meta name="keywords" content="math, maths, mathematics, school, homework, education">
|
||
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes">
|
||
<meta name="HandheldFriendly" content="true">
|
||
<meta name="referrer" content="always">
|
||
<link rel="preload" href="../images/style/font-champ-bold.ttf" as="font" type="font/ttf" crossorigin="">
|
||
<link rel="preload" href="../style4.css" as="style">
|
||
<link rel="preload" href="../main4.js" as="script">
|
||
<link rel="stylesheet" href="../style4.css">
|
||
<script src="../main4.js" defer="defer"></script>
|
||
<!-- Global site tag (gtag.js) - Google Analytics -->
|
||
<script async="" src="https://www.googletagmanager.com/gtag/js?id=UA-29771508-1"></script>
|
||
<script>
|
||
window.dataLayer = window.dataLayer || [];
|
||
function gtag(){dataLayer.push(arguments);}
|
||
gtag('js', new Date());
|
||
gtag('config', 'UA-29771508-1');
|
||
</script>
|
||
</head>
|
||
|
||
<body id="bodybg">
|
||
|
||
<div id="stt"></div>
|
||
<div id="adTop"></div>
|
||
<header>
|
||
<div id="hdr"></div>
|
||
<div id="tran"></div>
|
||
<div id="adHide"></div>
|
||
<div id="cookOK"></div>
|
||
</header>
|
||
|
||
<div class="mid">
|
||
|
||
<nav>
|
||
<div id="menuWide" class="menu"></div>
|
||
<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun"></a></div>
|
||
|
||
<div id="search" role="search"></div>
|
||
<div id="linkto"></div>
|
||
|
||
<div id="menuSlim" class="menu"></div>
|
||
<div id="menuTiny" class="menu"></div>
|
||
</nav>
|
||
|
||
<div id="extra"></div>
|
||
|
||
<article id="content" role="main">
|
||
|
||
<!-- #BeginEditable "Body" -->
|
||
|
||
|
||
<h1 class="center">Scalars and Vectors</h1>
|
||
|
||
<p class="center"><i>(... and Matrices)</i></p>
|
||
<p>What are Scalars and Vectors?</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>A <b>scalar</b> has only <b>magnitude</b> (size):</h3>
|
||
<p class="center">3.044, −7 and 2½ are scalars</p>
|
||
<p><a href="../measure/metric-length.html">Distance</a>, <a href="../measure/speed-velocity.html">speed</a>, <a href="../time.html">time</a>, <a href="../temperature-conversion.html">temperature</a>, <a href="../measure/metric-mass.html">mass</a>, <a href="../measure/metric-length.html">length</a>, <a href="../measure/metric-area.html">area</a>, <a href="../measure/metric-volume.html">volume</a>, density, charge, pressure, <a href="../physics/energy-work.html">energy</a>, <a href="../physics/energy-work.html">work</a> and <a href="../physics/power.html">power</a> are all scalars.</p>
|
||
</div>
|
||
|
||
<div class="example">
|
||
|
||
<h3>A <b>vector</b> has <b>magnitude</b> and <b>direction</b>:</h3>
|
||
<p class="center"><img src="images/vector-mag-dir.svg" alt="vector magnitude and direction" height="137" width="267"></p> <a href="../measure/metric-length.html">Displacement</a>, <a href="../measure/metric-speed.html">velocity</a>, <a href="../measure/metric-acceleration.html">acceleration</a>, <a href="../physics/force.html">force</a> and <a href="../physics/momentum.html">momentum</a> are all <a href="vectors.html">vectors</a>.</div>
|
||
<div class="simple"></div>
|
||
<p>And watch out for these special words:</p>
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="../measure/images/displace-distance.svg" alt="displacement vs distance" height="249" width="360"></p>
|
||
|
||
<h3>Distance vs Displacement</h3>
|
||
<ul>
|
||
<li>Distance is a scalar ("3 km")</li>
|
||
<li>Displacement is a vector ("3 km Southeast")</li>
|
||
</ul>
|
||
<p>You can walk a long distance, but your displacement may be small (or zero if you return to the start).</p>
|
||
<div style="clear:both"></div>
|
||
|
||
<h3>Speed vs Velocity</h3>
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="../images/art/ariel-run-ball.jpg" alt="dog run ball" height="134" width="180"></p>
|
||
<ul>
|
||
<li>Speed is how fast something moves.</li>
|
||
<li>Velocity is speed with a <b>direction</b>.</li>
|
||
</ul>
|
||
<p>Saying Ariel the Dog runs at <b>9 km/h</b> (kilometers per hour) is a speed.</p>
|
||
<p>But saying he runs <b>9 km/h Westwards</b> is a velocity.</p>
|
||
<p>See <a href="../measure/speed-velocity.html">Speed and Velocity</a> to learn more.</p>
|
||
|
||
|
||
<h2>Notation</h2>
|
||
|
||
<p>A vector is often written in <b>bold</b>, like <b>a</b> or <b>b</b> so we know it is not a scalar:</p>
|
||
<ul>
|
||
<li>so <b>c</b> is a vector, it has magnitude and direction</li>
|
||
<li>but c is a scalar, like 3 or 12.4</li>
|
||
</ul>
|
||
|
||
<div class="example">
|
||
<p>Example: k<b>b</b> is actually the scalar k times the vector <b>b</b>.</p>
|
||
</div>
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/vector-notation.svg" alt="vector notation" height="72" width="170"></p>
|
||
<p>A vector can also be written as the letters of its head and tail with an arrow above it, like this:</p>
|
||
|
||
|
||
<h2>Using Scalars</h2>
|
||
|
||
<p>Scalars are easy to use. Just treat them as normal numbers.</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: 3 kg + 4 kg = 7 kg</h3> </div>
|
||
|
||
|
||
<h2>Using Vectors</h2>
|
||
|
||
<p>The page on <a href="vectors.html">vectors</a> has more detail, but here is a quick summary:</p>
|
||
<p>We can add two vectors by joining them head-to-tail:</p>
|
||
<p class="center"><img src="images/vector-add.svg" alt="vector addition" height="89" width="244"></p>
|
||
<p>We can subtract one vector from another:</p>
|
||
<ul>
|
||
<li>first we reverse the direction of the vector we want to subtract,</li>
|
||
<li>then add them as usual:</li>
|
||
</ul>
|
||
<p class="center"><img src="images/vector-subtract.gif" alt="vector subtraction" height="115" width="264"><br>
|
||
<span class="large"><b>a</b> − <b>b</b></span></p>
|
||
<p>We can multiply a vector by a scalar (called "scaling" a vector):</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: multiply the vector <b>m</b> = (7,3) by the scalar 3</h3>
|
||
|
||
<table style="border: 0;">
|
||
<tbody>
|
||
<tr>
|
||
<td><img src="images/vector-scaling.gif" alt="vector scaling" height="135" width="184"></td>
|
||
<td> </td>
|
||
<td><b>a</b> = 3<b>m</b> = (3×7,3×3) = (21,9)</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>It still points in the same direction, but is 3 times longer</p>
|
||
</div>
|
||
<p>(And now you know why numbers are called "scalars", because they "scale" the vector up or down.)</p>
|
||
|
||
|
||
<h2>Polar or Cartesian</h2>
|
||
|
||
<p>A vector can be in:</p>
|
||
<ul>
|
||
<li><b>magnitude and direction</b> (Polar) form,</li>
|
||
<li>or in <b>x and y</b> (Cartesian) form</li>
|
||
</ul>
|
||
<p>Like this:</p>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td style="text-align:center;"><img src="images/vector-polar.svg" alt="vector polar" height="171" width="190"></td>
|
||
<td style="text-align:center;"><=></td>
|
||
<td style="text-align:center;"><img src="images/vector-cartesian.svg" alt="vector cartesian" height="171" width="188"></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;">Vector <b>a</b> in Polar<br>
|
||
Coordinates</td>
|
||
<td style="text-align:center;"> </td>
|
||
<td style="text-align:center;">Vector <b>a</b> in Cartesian<br>
|
||
Coordinates</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p class="center">(Read how to convert them at <a href="../polar-cartesian-coordinates.html">Polar and Cartesian Coordinates</a>.)</p>
|
||
|
||
<div class="example">
|
||
|
||
<h3>Example: the vector <b>13 at 22.6°</b></h3>
|
||
<p>In Polar (magnitude and direction) form:</p>
|
||
<p class="center"><img src="../geometry/images/coordinates-polar.svg" alt="coordinates polar 13 at 22.6 degrees" height="232" width="348"><br>
|
||
The vector <b>13 at 22.6°</b></p>
|
||
<p>Is approximately <b>(12,5)</b> In Cartesian (x,y) form:</p>
|
||
<p class="center"><img src="../geometry/images/coordinates-cartesian-vector.svg" alt="coordinates polar 13 at 22.6 degrees" height="232" width="348"><br>
|
||
The vector <b>(12,5)</b></p>
|
||
</div>
|
||
<p>Have a try of the <a href="vector-calculator.html">Vector Calculator</a> to get a feel for how it all works.</p>
|
||
|
||
|
||
<h2>Multiplying a Vector by a Vector (Dot Product and Cross Product)</h2>
|
||
|
||
<table width="100%" border="0">
|
||
<tbody>
|
||
<tr>
|
||
<td><img src="images/dot-product-1.gif" alt="dot product 1" height="139" width="164"></td>
|
||
<td>
|
||
<p>How do we <b>multiply two vectors</b> together? There is more than one way!</p>
|
||
<ul>
|
||
<li>The scalar or <a href="vectors-dot-product.html">Dot Product</a> (the result is a scalar).</li>
|
||
<li>The vector or <a href="vectors-cross-product.html">Cross Product</a> (the result is a vector).</li>
|
||
</ul>
|
||
<p>(Read those pages for more details.)</p>
|
||
</td>
|
||
</tr>
|
||
</tbody></table>
|
||
|
||
|
||
<h2>More Than 2 Dimensions</h2>
|
||
|
||
<p>Vectors also work perfectly well in 3 or more dimensions:</p>
|
||
<p class="center"><img src="images/vector-3da.svg" alt="vector 3d: (1,4,5)" height="270" width="320"><br>
|
||
<b>The vector (1,4,5)</b></p>
|
||
|
||
|
||
<h2>List of Numbers</h2>
|
||
|
||
<p>So a vector can be thought of as a<b> list numbers</b>:</p>
|
||
<ul>
|
||
<li>2 numbers for 2D space, such as (4,7)</li>
|
||
<li>3 numbers for 3D space, such as (1,4,5)</li>
|
||
<li>etc</li>
|
||
</ul>
|
||
<p> </p>
|
||
<div class="fun">
|
||
|
||
|
||
<h2>Scalars, Vectors and Matrices</h2>
|
||
|
||
<p>And when we include <a href="matrix-introduction.html">matrices</a> we get this interesting pattern:</p>
|
||
<ul>
|
||
<div class="bigul">
|
||
<li>A <b>scalar</b> is a number, like <b>3, -5, 0.368, etc</b>,</li>
|
||
<li>A <b>vector</b> is a <b>list</b> of numbers (can be in a row or column),</li>
|
||
<li>A <b>matrix</b> is an <b>array</b> of numbers (one or more rows, one or more columns).</li>
|
||
</div>
|
||
</ul>
|
||
<p class="center"><img src="images/scalar-vector-matrix.svg" alt="scalar vector matrix" height="173" width="485"></p>
|
||
<p>In fact a <b>vector is also a matrix</b>! Because a matrix can have just one row or one column.</p>
|
||
<p>So the rules that work for matrices also work for vectors.</p>
|
||
</div>
|
||
<p> </p>
|
||
<div class="questions">11913, 11914, 11915, 11916, 11917, 11918, 11919, 11920, 11921, 11922</div>
|
||
|
||
<div class="related">
|
||
<a href="vector-calculator.html">Vector Calculator</a>
|
||
<a href="vectors-dot-product.html">Dot Product</a>
|
||
<a href="vectors-cross-product.html">Cross Product</a>
|
||
<a href="vector-unit.html">Unit Vector</a>
|
||
<a href="index.html">Algebra Index</a>
|
||
</div>
|
||
<!-- #EndEditable -->
|
||
|
||
</article>
|
||
|
||
<div id="adend" class="centerfull noprint"></div>
|
||
<footer id="footer" class="centerfull noprint"></footer>
|
||
<div id="copyrt">Copyright © 2022 Rod Pierce</div>
|
||
|
||
</div>
|
||
|
||
|
||
|
||
</body><!-- #EndTemplate -->
|
||
<!-- Mirrored from www.mathsisfun.com/algebra/scalar-vector-matrix.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:49:36 GMT -->
|
||
</html> |