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434 lines
18 KiB
HTML
434 lines
18 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<!-- Mirrored from www.mathsisfun.com/data/cartesian-coordinates.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:43:49 GMT -->
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<meta http-equiv="content-type" content="text/html; charset=UTF-8">
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<title>Cartesian Coordinates</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Cartesian Coordinates</h1>
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<p class="center">Cartesian coordinates can be used to pinpoint where we are on a map or graph.</p>
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<h2>Cartesian Coordinates</h2>
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<p>Using Cartesian Coordinates we mark a <a href="../geometry/point.html">point</a> on a graph by <b>how far along</b> and <b>how far up</b> it is:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="text-align:center;"><img src="../geometry/images/coordinates-cartesian.svg" alt="graph with point (12,5)" height="232" width="348"></td>
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</tr>
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<tr>
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<td class="larger" align="center">
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The point <b>(12,5)</b> is 12 units along, and 5 units up.</td>
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</tr>
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</tbody></table><p> </p>
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<div class="words">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../geometry/images/coordinates-rectangular.svg" alt="graph with point (12,5)" height="116" width="174"></p>
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<p>They are also called <b>Rectangular Coordinates</b> because it is like we are forming a rectangle.</p>
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<div style="clear:both"></div>
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</div>
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<h2>X and Y Axis</h2>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><img src="../geometry/images/x-axis.svg" alt="x axis" height="36" width="91"></td>
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<td>The <i>left-right</i> (<b>horizontal</b>) direction is commonly called <b>X</b>.</td>
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</tr>
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<tr>
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<td style="text-align:center;"><img src="../geometry/images/y-axis.svg" alt="y axis" height="71" width="36"></td>
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<td>The <i>up-down</i> (<b>vertical</b>) direction is commonly called <b>Y</b>.</td>
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</tr>
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</tbody></table>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td colspan="2" valign="middle">Put them together on a graph ...</td>
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</tr>
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<tr>
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<td valign="middle">
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<p><img src="../geometry/images/x-y-axes.svg" alt="x y axes" height="203" width="291"></p>
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</td>
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<td>
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<p> </p>
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<p align="right">... and we are ready to go</p></td>
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</tr>
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<tr>
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<td colspan="2">
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<p>Where they cross over is the "0" point, <b><br>
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we measure everything from there</b>.</p>
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<ul>
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<li><span style="color:#0000ff">The <b>X Axis</b> runs horizontally through zero</span></li>
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<li><span style="color:#ff0000">The <b>Y Axis</b> runs vertically through zero</span></li>
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</ul> </td>
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</tr>
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</tbody></table>
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<div class="words">
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<p><b>Axis</b>: The reference line from which distances
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are measured.</p>
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<p>The plural of Axis is <b><i>Axes</i></b>, and is pronounced <i>ax-eez</i></p>
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</div>
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<p> </p>
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<div class="example">
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<h3>Example:</h3>
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<p><img src="../geometry/images/coord-1-quadrant.svg" alt="coord (6,4) " style="float:left; margin: 10px;" height="132" width="181"></p>
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<p>Point <b>(6,4)</b> is</p>
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<p>6 units across (in the <b>x</b> direction), and</p>
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<p>4 units up (in the <b>y</b> direction)</p>
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<p> </p>
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<p>So <b>(6,4)</b> means:</p>
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<p class="center larger">Go along 6 and then go up 4 then "plot the dot".</p>
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<div style="clear:both"></div>
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</div>
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<p>And you can remember which axis is which by:</p>
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<p class="center large">x is A CROSS, so x is ACROSS the page.</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td><a href="cartesian-coordinates-interactive.html"><img src="images/interactive-cartesian-coordinates.gif" alt="interactive-cartesian-coordinates" height="150" width="150"></a></td>
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<td> </td>
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<td>
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<h2>Play With It !</h2>
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<p>Now is a good time to play with</p>
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<p class="center"><a href="cartesian-coordinates-interactive.html">Interactive Cartesian Coordinates</a></p>
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<p>to see for yourself how it all works.</p></td>
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</tr>
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</tbody></table>
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<h2>Like 2 Number Lines Put Together</h2>
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<p>It is like we put two <a href="../number-line.html">Number Lines</a> together, one going left-right, and the other going down-up.</p>
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<img src="images/two-number-lines-cartesian.svg" alt="Two Number Lines make cartesian coords" style="max-" height="263" width="695">
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<h2>Direction</h2>
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<table align="center" width="90%" border="0">
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<tbody>
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<tr>
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<td><img src="../images/style/right.svg" alt="right arrow" height="46" width="46"></td>
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<td>
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<p>As <b>x</b> increases, the point moves further <b>right</b>.<br>
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When x decreases, the point moves further to the left.</p></td>
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</tr>
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<tr>
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<td><img src="../images/style/up.svg" alt="up arrow" height="46" width="46"></td>
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<td>
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<p>As <b>y</b> increases, the point moves further <b>up</b>.<br>
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When y decreases, the point moves further down.</p></td>
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</tr>
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</tbody></table>
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<h2>Writing Coordinates</h2>
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<p>The coordinates are always written in a certain order:</p>
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<ul>
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<li>the horizontal distance first,</li>
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<li>then the vertical distance.</li>
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</ul>
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<p>This is called an "<b>ordered pair</b>" (a <b>pair</b> of numbers in a special <b>order</b>)</p>
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<p>And usually the numbers are separated by a comma, and parentheses are put around the whole thing like this:</p>
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<p class="huge" align="center">(3,2)</p>
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<div class="example">
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<p><b>Example: (3,2) means 3 units to the right, and 2 units up</b></p>
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</div>
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<div class="example">
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<p><b>Example: (0,5) means 0 units to the right, and 5 units up.</b></p>
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<p>In other words, only 5 units up.</p>
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</div>
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<h2>The Origin</h2>
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<p>The point (0,0) is given the special name "The Origin", and is sometimes given the letter "O".</p>
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<h2>Abscissa and Ordinate</h2>
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<p>You may hear the words "Abscissa" and "Ordinate" ... they are just the <span class="large">x</span> and <span class="large">y</span> values:</p>
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<ul>
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<li>Abscissa: the horizontal ("x") value in a pair of coordinates: how far <b>along</b> the point is</li>
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<li>Ordinate: the vertical ("y") value in a pair of coordinates: how far <b>up or down</b> the point is</li>
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</ul>
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<h2>"Cartesian" ... ?</h2>
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<div class="words">
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<p>They are called <i>Cartesian</i> because the idea was developed by the mathematician and philosopher <b>Rene Descartes</b> who was also known as <b><i>Cartesius</i></b>.</p>
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<p>He is also famous for saying <i>"I think, therefore I am"</i>.</p>
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</div>
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<h2>What About Negative Values of X and Y?</h2>
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<p>Just like with the Number Line, we can also have negative values.</p>
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<p class="center larger">Negative: start at zero and <b>head in the opposite direction</b>:</p>
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<ul>
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<li>Negative x goes <b>to the left</b></li>
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<li>Negative y goes <b>down</b></li>
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</ul>
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<div class="simple">
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<table align="center" cellspacing="4" cellpadding="5" border="0">
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<tbody>
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<tr>
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<td style="text-align:center;"><img src="../geometry/images/coords-4-quadrants.svg" alt="coords in 4 quadrants" height="249" width="339"></td>
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<td style="text-align:left;">
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<p>So, for a <font color="red">negative</font> number:</p>
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<ul>
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<li><span class="larger">go <font color="red">backwards</font> for
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x </span></li>
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<li><span class="larger">go <font color="red">down</font> for y </span></li>
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</ul>
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<p><br>
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<font color="purple">For example <b>(-6,4)</b> means:</font><br>
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go <b>back</b> along the x axis 6 then go up 4.</p>
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<p><br>
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<font color="purple">And <b>(-6,-4)</b> means:</font><br>
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go <b>back</b> along the x axis 6 then go <b>down 4.</b></p></td>
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</tr>
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</tbody></table>
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</div>
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<p> </p>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td> </td>
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<td>
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<h2>Four Quadrants</h2>
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<p>When we include negative values, the x and y axes divide the space up into 4 pieces:</p>
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<p class="center"><b>Quadrants I, II, III</b> and<b> IV</b></p>
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<p><i>(They are numbered in a <a href="../geometry/clockwise-counterclockwise.html">counterclockwise</a> direction)</i></p></td>
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<td><img src="../geometry/images/cartesian-quadrants.svg" alt="Quadrants" height="191" width="257"></td>
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</tr>
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</tbody></table>
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<div class="indent50px">
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<p>In <b>Quadrant I</b> both x and y are positive, but ...</p>
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<ul>
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<li>in <b>Quadrant II</b> x is <span style="color: #ff0000;">negative</span> (y is still positive),</li>
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<li>in <b>Quadrant III</b> both x and y are <span style="color: #ff0000;">negative</span>, and</li>
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<li>in <b>Quadrant IV</b> x is positive again, while y is <span style="color: #ff0000;">negative</span>.</li>
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</ul>
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</div>
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<p>Like this:</p>
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<div class="beach">
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<table style="border: 0;">
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<tbody>
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<tr>
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<td><img src="images/cartesian-coordinates.gif" alt="cartesian coordinates" height="286" width="284"></td>
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<td>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr style="text-align:center;">
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<th width="100">Quadrant</th>
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<th width="120">X<br>
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(horizontal)</th>
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<th width="120">Y<br>
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(vertical)</th>
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<th width="100">Example</th>
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</tr>
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<tr style="text-align:center;">
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<td class="larger" width="100"><b>I</b></td>
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<td style="width:120px;">Positive</td>
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<td style="width:120px;">Positive</td>
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<td style="width:100px;">(3,2)</td>
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</tr>
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<tr style="text-align:center;">
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<td class="larger" width="100"><b>II</b></td>
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<td style="width:120px;"><i>Negative</i></td>
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<td style="width:120px;">Positive</td>
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<td style="width:100px;"> </td>
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</tr>
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<tr style="text-align:center;">
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<td class="larger" width="100"><b>III</b></td>
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<td style="width:120px;"><i>Negative</i></td>
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<td style="width:120px;"><i>Negative</i></td>
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<td style="width:100px;">(-2,-1)</td>
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</tr>
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<tr style="text-align:center;">
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<td class="larger" width="100"><b>IV</b></td>
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<td style="width:120px;">Positive</td>
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<td style="width:120px;"><i>Negative</i></td>
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<td style="width:100px;"> </td>
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</tr>
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</tbody></table></td>
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</tr>
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</tbody></table>
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</div>
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<p> </p>
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<div class="example">
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<p>Example: The point "A" (3,2) is 3 units along, and 2 units up.</p>
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<p>Both x and y are positive, so that point is in "Quadrant I"</p>
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</div>
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<div class="example">
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<p>Example: The point "C" (-2,-1) is 2 units along in the negative direction, and 1 unit down (i.e. negative direction).</p>
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<p>Both x and y are negative, so that point is in "Quadrant III"</p>
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</div>
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<p> </p>
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<div class="words">
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Note: The word <b>Quadrant</b> comes form <i>quad</i> meaning <b>four</b>. For example, four babies born at one birth are called <i>quadruplets</i>, a four-legged animal is a <i>quadruped</i>. and a <i>quadrilateral</i> is a four-sided polygon.
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</div>
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<h2>Dimensions: 1, 2, 3 and more ...</h2>
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<p>Think about this:</p>
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<div class="simple">
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<table align="center" width="87%" border="0">
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<tbody>
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<tr>
|
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<th class="huge" align="center" width="12%">1</th>
|
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<td width="88%">
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<p>The <a href="../number-line.html">Number Line</a> can only go:</p>
|
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<ul>
|
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<li>left-right</li>
|
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</ul> <p>so any position needs just <b>one number</b></p></td>
|
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</tr>
|
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<tr>
|
|
<th class="huge" align="center" width="12%">2</th>
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|
|
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<td width="88%">
|
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<p>Cartesian coordinates can go:</p>
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<ul>
|
|
<li>left-right, and</li>
|
|
<li>up-down</li>
|
|
</ul> <p>so any position needs <b>two numbers</b></p></td>
|
|
</tr>
|
|
<tr>
|
|
<th class="huge" align="center" width="12%">3</th>
|
|
|
|
<td width="88%">
|
|
<p>How do we locate a spot in the real world (such as the tip of your nose)? We need to know:</p>
|
|
<ul>
|
|
<li>left-right,</li>
|
|
<li>up-down, and</li>
|
|
<li>forward-backward,</li>
|
|
</ul> <p>that is <b>three numbers</b>, or 3 dimensions!</p></td>
|
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</tr>
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</tbody></table>
|
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</div>
|
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|
|
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<h2>3 Dimensions</h2>
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|
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<p>Cartesian coordinates can be used for locating points in 3 dimensions as in this example:</p>
|
|
|
|
<table style="border: 0; margin:auto;">
|
|
<tbody>
|
|
<tr>
|
|
<td style="text-align:center;"><img src="../geometry/images/cartesian-3d.svg" alt="cartesian in 3d" height="250" width="310"></td>
|
|
</tr>
|
|
<tr>
|
|
<td class="larger" align="center">
|
|
Here the point <b>(2, 4, 5)</b> is shown in<br>
|
|
three-dimensional Cartesian coordinates.<br>
|
|
</td>
|
|
</tr>
|
|
</tbody></table>
|
|
<p>In fact, this idea can be continued into four dimensions and more - I just can't work out how to illustrate that for you!</p>
|
|
<p> </p>
|
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<div class="questions">753,754,755,756,1142,1143,1144, 3208, 3209, 3210</div>
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<div class="related">
|
|
<a href="cartesian-coordinates-interactive.html">Interactive Cartesian Coordinates</a>
|
|
<a href="click-coordinate.html">Hit the Coordinate Game</a>
|
|
<a href="../polar-cartesian-coordinates.html">Polar and Cartesian Coordinates</a>
|
|
<a href="../number-line.html">The Number Line</a>
|
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<a href="graphs-index.html">Graphs Index</a>
|
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</div>
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