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319 lines
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HTML
319 lines
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Vectors</h1>
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<p class="center">This is a vector:</p>
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<p class="center"><img src="images/vector.gif" width="142" height="81" alt="vector" /></p>
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<p>A vector has <b>magnitude</b> (size) and <b>direction</b>:</p>
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<p class="center"><img src="images/vector-mag-dir.svg" alt="vector magnitude and direction" /></p>
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<p class="center">The length of the line shows its magnitude and the arrowhead points in the direction.</p>
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<p>We can add two vectors by joining them head-to-tail:</p>
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<p class="center"><img src="images/vector-add.svg" width="248" height="92" alt="vector add a+b" /></p>
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<p>And it doesn't matter which order we add them, we get the same result:</p>
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<p class="center"><img src="images/vector-add2.gif" width="247" height="95" alt="vector add b+a" /></p>
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<div class="example">
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<h3>Example: A plane is flying along, pointing North, but there is a wind coming from the North-West.</h3>
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<p class="center"><img src="images/vector-airplane.svg" alt="vector airplane, propellor and wind" /></p>
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<p>The two vectors (the velocity caused by the propeller, and the velocity of the wind) result in a slightly slower ground speed heading a little East of North.</p>
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<p>If you watched the plane from the ground it would seem to be slipping sideways a little.</p>
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<p class="center"><img src="images/vector-airplane2.gif" width="134" height="154" alt="vector airplane ahead and slightly sideways" /></p>
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<p>Have you ever seen that happen? Maybe you have seen birds struggling against a strong wind that seem to fly sideways. Vectors help explain that.</p>
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</div>
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<p><a href="../measure/metric-speed.html">Velocity</a>, <a href="../measure/metric-acceleration.html">acceleration</a>, <a href="../physics/force.html">force</a> and many other things are vectors.</p>
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<h2>Subtracting</h2>
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<p>We can also subtract one vector from another:</p>
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<ul>
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<li>first we reverse the direction of the vector we want to subtract,</li>
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<li>then add them as usual:</li>
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</ul>
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<p class="center"><img src="images/vector-subtract.gif" width="264" height="115" alt="vector subtract a-b = a + (-b)" /><br>
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<b>a</b> − <b>b</b></p>
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<h2>Notation</h2>
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<p>A vector is often written in <b>bold</b>, like <b>a</b> or <b>b</b>.</p>
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<table style="border: 0; margin:auto;">
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<tr>
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<td>A vector can also be written as the letters<br>
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of its head and tail with an arrow above it, like this:</td>
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<td> </td>
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<td><img src="images/vector-notation.svg" alt="vector notation a=AB, head, tail" /></td>
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</tr>
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</table>
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<h2>Calculations</h2>
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<p>Now ... how do we do the calculations?</p>
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<p>The most common way is to first break up vectors into x and y parts, like this:</p>
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<p class="center"><img src="images/vector-xy-components.gif" width="122" height="92" alt="vector xy components" /></p>
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<p class="center">The vector <b>a</b> is broken up into<br>
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the two vectors <b>a<sub>x</sub></b> and <b>a<sub>y</sub></b></p>
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<p class="center">(We <a href="#magdir">see later</a> how to do this.)</p>
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<h2>Adding Vectors</h2>
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<p>We can then add vectors by <b>adding the x parts</b> and <b>adding the y parts</b>:</p>
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<p class="center"><img src="images/vector-add3.gif" width="484" height="130" alt="vector add example" /></p>
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<p class="center">The vector (8, 13) and the vector (26, 7) add up to the vector (34, 20)</p>
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<div class="example">
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<h3>Example: add the vectors <b>a</b> = (8, 13) and <b>b</b> = (26, 7)</h3>
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<p><b>c</b> = <b>a</b> + <b>b</b></p>
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<p><b>c</b> = (8, 13) + (26, 7) = (8+26, 13+7) = (34, 20)</p>
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</div>
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<p>When we break up a vector like that, each part is called a <b>component</b>:</p>
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<link rel="stylesheet" type="text/css" href="../stylejs.css"><script src="../geometry/images/geom-vector.js"></script>
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<script>geomvectorMain('xy');</script>
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<h2>Subtracting Vectors</h2>
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<p>To subtract, first reverse the vector we want to subtract, then add.</p>
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<div class="example">
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<h3>Example: subtract <b>k</b> = (4, 5) from <b>v</b> = (12, 2)</h3>
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<p><b>a</b> = <b>v</b> + −<b>k</b></p>
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<p><b>a</b> = (12, 2) + −(4, 5) = (12, 2) + (−4, −5) = (12−4, 2−5) = (8, −3)</p>
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</div>
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<h2>Magnitude of a Vector</h2>
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<p>The magnitude of a vector is shown by two vertical bars on either side of the vector:</p>
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<p class="large center">|<b>a</b>|</p>
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<p>OR it can be written with double vertical bars (so as not to confuse it with absolute value):</p>
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<p class="large center">||<b>a</b>||</p>
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<p>We use <a href="../pythagoras.html">Pythagoras' theorem</a> to calculate it:</p>
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<p class="large center">|<b>a</b>| = √( x<sup>2</sup> + y<sup>2</sup> )</p>
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<div class="example">
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<h3>Example: what is the magnitude of the vector <b>b</b> = (6, 8) ?</h3>
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<p>|<b>b</b>| = √( 6<sup>2</sup> + 8<sup>2</sup>) = √( 36+64) = √100 = 10</p>
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</div>
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<p>A vector with magnitude 1 is called a <a href="vector-unit.html">Unit Vector</a>.</p>
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<h2>Vector vs Scalar</h2>
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<p>A <b>scalar</b> has <b>magnitude</b> (size) <b>only</b>.</p>
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<div class="def">
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<p>Scalar: just a number (like 7 or −0.32) ... definitely not a vector.</p>
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</div>
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<p>A <b>vector</b> has <b>magnitude and direction</b>, and is often written in <b>bold</b>, so we know it is not a scalar:</p>
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<ul>
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<li>so <b>c</b> is a vector, it has magnitude and direction</li>
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<li>but c is just a value, like 3 or 12.4</li>
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</ul>
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<div class="example">
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<h3>Example: k<b>b</b> is actually the scalar k times the vector <b>b</b>.</h3>
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</div>
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<h2>Multiplying a Vector by a Scalar</h2>
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<p>When we multiply a vector by a scalar it is called "scaling" a vector, because we change how big or small the vector is.</p>
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<div class="example">
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<h3>Example: multiply the vector <b>m</b> = (7, 3) by the scalar 3</h3>
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<table style="border: 0;">
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<tr>
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<td><img src="images/vector-scaling.gif" width="184" height="135" alt="vector scaling" /></td>
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<td> </td>
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<td><b>a</b> = 3<b>m</b> = (3×7, 3×3) = (21, 9)</td>
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</tr>
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</table>
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<p>It still points in the same direction, but is 3 times longer</p>
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</div>
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<p>(And now you know why numbers are called "scalars", because they "scale" the vector up or down.)</p>
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<p> </p>
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<h2>Multiplying a Vector by a Vector (Dot Product and Cross Product)</h2>
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<table width="100%" border="0">
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<tr>
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<td><img src="images/dot-product-1.gif" width="164" height="139" alt="dot product magnitude and angle" /></td>
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<td>
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<p>How do we <b>multiply two vectors</b> together? There is more than one way!</p>
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<ul>
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<li>The scalar or <a href="vectors-dot-product.html">Dot Product</a> (the result is a scalar).</li>
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<li>The vector or <a href="vectors-cross-product.html">Cross Product</a> (the result is a vector).</li>
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</ul>
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<p>(Read those pages for more details.)</p>
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</td>
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</tr>
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</table>
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<p> </p>
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<h2>More Than 2 Dimensions</h2>
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<p>Vectors also work perfectly well in 3 or more dimensions:</p>
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<p class="center"><img src="images/vector-3da.svg" width="320" height="270" alt="vector in 3d" /><br>
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<b>The vector (1, 4, 5)</b></p>
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<div class="example">
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<h3>Example: add the vectors <b>a</b> = (3, 7, 4) and <b>b</b> = (2, 9, 11)</h3>
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<p><b>c</b> = <b>a</b> + <b>b</b></p>
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<p><b>c</b> = (3, 7, 4) + (2, 9, 11) = (3+2, 7+9, 4+11) = (5, 16, 15)</p>
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</div>
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<div class="example">
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<h3>Example: what is the magnitude of the vector <b>w</b> = (1, −2, 3) ?</h3>
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<p class="center">|<b>w</b>| = √( 1<sup>2</sup> + (−2)<sup>2 </sup> + 3<sup>2 </sup>) = √( 1+4+9) = √14</p>
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</div>
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<p>Here is an example with 4 dimensions (but it is hard to draw!):</p>
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<div class="example">
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<h3>Example: subtract (1, 2, 3, 4) from (3, 3, 3, 3)</h3>
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<p class="center">(3, 3, 3, 3) + −(1, 2, 3, 4)<br>
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= (3, 3, 3, 3) + (−1,−2,−3,−4)<br>
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= (3−1, 3−2, 3−3, 3−4)<br>
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= (2, 1, 0, −1)</p>
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</div>
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<p> </p>
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<h2><a id="magdir"></a>Magnitude and Direction</h2>
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<p>We may know a vector's magnitude and direction, but want its x and y lengths (or vice versa):</p>
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<table style="border: 0; margin:auto;">
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<tr>
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<td style="text-align:center;"><img src="images/vector-polar.svg" alt="vector polar" /></td>
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<td style="text-align:center;"><=></td>
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<td style="text-align:center;"><img src="images/vector-cartesian.svg" alt="vector cartesian" /></td>
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</tr>
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<tr>
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<td style="text-align:center;">Vector <b>a</b> in Polar<br>
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Coordinates</td>
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<td style="text-align:center;"> </td>
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<td style="text-align:center;">Vector <b>a</b> in Cartesian<br>
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Coordinates</td>
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</tr>
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</table>
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<p>You can read how to convert them at <a href="../polar-cartesian-coordinates.html">Polar and Cartesian Coordinates</a>, but here is a quick summary:</p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tr>
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<th>From Polar Coordinates (r,<i>θ</i>)<br>
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to Cartesian Coordinates (x,y)</th>
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<td> </td>
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<th><span class="larger">From Cartesian Coordinates (x,y)<br>
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to Polar Coordinates (r,θ)</span></th>
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</tr>
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<tr>
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<td>
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<ul>
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<li><b>x = r</b> × <b>cos( <i>θ</i> )</b></li>
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<li><b>y = r</b> × <b>sin(<i> θ</i> )</b></li>
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</ul>
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</td>
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<td> </td>
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<td>
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<ul>
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<li><b>r = √ ( x<sup>2</sup> + y<sup>2 </sup>)</b></li>
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<li><b><i>θ</i> = tan<sup>-1 </sup>( y / x )</b></li>
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</ul>
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</td>
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</tr>
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</table>
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</div>
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<p> </p>
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<p> </p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/vector-ex1c.svg" alt="vector example two people pull" /></p>
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<h2>An Example</h2>
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<p>Sam and Alex are pulling a box.</p>
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<ul>
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<li>Sam pulls with 200 Newtons of force at 60°</li>
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<li>Alex pulls with 120 Newtons of force at 45° as shown</li>
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</ul>
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<p>What is the combined <a href="../physics/force.html">force</a>, and its direction?</p>
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<p> </p>
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<p>Let us add the two vectors head to tail:</p>
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<p class="center"><img src="images/vector-ex1a.gif" width="176" height="146" alt="vectors: angles and magnitudes" /></p>
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<p>First convert from polar to Cartesian (to 2 decimals):</p>
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<p>Sam's Vector:</p>
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<ul>
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<li><b>x = r × cos( <i>θ</i> ) = 200 × cos(60°) = 200 × 0.5 = 100</b></li>
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<li><b>y = r × sin(<i> θ</i> ) = 200 × sin(60°) = 200 × 0.8660 = 173.21</b></li>
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</ul>
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<p>Alex's Vector:</p>
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<ul>
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<li><b>x = r × cos( <i>θ</i> ) = 120 × cos(−45°) = 120 × 0.7071 = 84.85</b></li>
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<li><b>y = r × sin(<i> θ</i> ) = 120 × sin(−45°) = 120 × -0.7071 = −84.85</b></li>
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</ul>
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<p>Now we have:</p>
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<p class="center"><img src="images/vector-ex1b.gif" width="190" height="137" alt="vectors: components" /></p>
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<p>Add them:</p>
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<p class="center larger">(100, 173.21) + (84.85, −84.85) = (184.85, 88.36)</p>
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<p>That answer is valid, but let's convert back to polar as the question was in polar:</p>
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<ul>
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<li><b>r = √ ( x<sup>2</sup> + y<sup>2 </sup>) = √ ( 184.85<sup>2</sup> + 88.36<sup>2 </sup>)</b> = <b> 204.88</b></li>
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<li><b><i>θ</i> = tan<sup>-1 </sup>( y / x ) = tan<sup>-1 </sup>( 88.36 / 184.85 ) = 25.5°</b></li>
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</ul>
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<p class="center"><span class="large">And we have this (rounded) result:</span><br>
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<img src="images/vector-ex1d.gif" width="138" height="140" alt="vector result" /></p>
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<p class="center"><span class="large">And it looks like this for Sam and Alex:</span><br>
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<img src="images/vector-ex1e.svg" alt="vector combined pull force" /></p>
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<p>They might get a better result if they were shoulder-to-shoulder!</p>
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<p> </p>
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<div class="related">
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<a href="vectors-dot-product.html">Dot Product</a>
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