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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/html4/strict.dtd"> <html> <head> <title>page_40</title> <link rel="stylesheet" href="reset.css" type="text/css" media="all"> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> </head> <body> <table summary="top nav" border="0" width="100%"> <tr> <td align="left" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_39.html">< previous page</a></td> <td id="ebook_previous" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_40</strong></td> <td align="right" width="30%" style="background: #EEF3E2"><a style="color: blue; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_41.html">next page ></a></td> </tr> <tr> <td id="ebook_page" align="left" colspan="3" style="background: #ffffff; padding: 20px;"> <table border="0" width="100%" cellpadding="0"><tr><td align="center"> <table border="0" cellpadding="2" cellspacing="0" width="100%"><tr><td align="left"></td> <td align="right"></td> </tr></table></td></tr><tr><td align="left"><p></p><table border="0" cellspacing="0" cellpadding="0" width="100%"><tr><td align="right"><font face="Times New Roman, Times, Serif" size="2" color="#FF0000">Page 40</font></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">where i,j, and k correspond to the unit vectors along the <i>a</i>-frame coordinate axes. With <i>b</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>4</sub></font><font face="Times New Roman, Times, Serif" size="3"> interpreted as the real component and <i>b</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>1</sub></font><font face="Times New Roman, Times, Serif" size="3">-<i>b</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>3</sub></font><font face="Times New Roman, Times, Serif" size="3"> as orthogonal imaginary components of b, the complex conjugate of b is</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="fd12b6296a88c43f4c02e9c53805a7a4.gif" border="0" alt="0040-01.GIF" width="190" height="21" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">A vector quantity v = [<i>v</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>1</sub></font><font face="Times New Roman, Times, Serif" size="3">, <i>v</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>2</sub></font><font face="Times New Roman, Times, Serif" size="3">, <i>v</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>3</sub></font><font face="Times New Roman, Times, Serif" size="3">] is expressed as the quaternion b</font><font face="Times New Roman, Times, Serif" size="1"><sub><i>v</i></sub></font><font face="Times New Roman, Times, Serif" size="3"> = (<i>v</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>1</sub></font><font face="Times New Roman, Times, Serif" size="3">, <i>v</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>2</sub></font><font face="Times New Roman, Times, Serif" size="3">, <i>v</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>3</sub></font><font face="Times New Roman, Times, Serif" size="3">, 0). The symbol 潞 is used to denote the quaternion product. The quaternion product has the following properties:</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="888ebe2bf7ffddb6448e4e69d9085cdc.gif" border="0" alt="0040-02.GIF" width="246" height="52" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">Therefore the quaternion product of b and c is</font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="2a43762c06d1ea1d0f869d87190904a4.gif" border="0" alt="0040-03.GIF" width="450" height="21" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="1232fa5ddc25d55108fe38a0ae26a18a.gif" border="0" alt="0040-04.GIF" width="399" height="40" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="38285b4f28b5a96cbf7099e8ea333853.gif" border="0" alt="0040-05.GIF" width="415" height="72" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="c785c92bc446abc583849802db258a76.gif" border="0" alt="0040-06.GIF" width="415" height="72" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3">When quaternions are used, the transformation of the vector quantity r</font><font face="Times New Roman, Times, Serif" size="2"><sup><i>a</i></sup></font><font face="Times New Roman, Times, Serif" size="3"> from frame <i>a</i> to frame <i>b</i> is</font><font face="Times New Roman, Times, Serif" size="3" color="#FFFF00"></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="13cca16a344646b112c9abab9af05a46.gif" border="0" alt="0040-07.GIF" width="461" height="21" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="bc7819966857fde7c3327e97c9c581ca.gif" border="0" alt="0040-08.GIF" width="442" height="72" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="12"></td> <td rowspan="5"></td></tr><tr><td colspan="3"></td></tr><tr><td></td> <td><font face="Times New Roman, Times, Serif" size="3"><img src="ef93eddf2c2b23ed94612fc572f01d79.gif" border="0" alt="0040-09.GIF" width="441" height="101" /></font></td><td></td></tr><tr><td colspan="3"></td></tr><tr><td colspan="3" height="1"></td></tr></table></td></tr></table><p><font size="0"></font></p>聽 </td> </tr> <tr> <td align="left" width="30%" style="background: #EEF3E2"><a style="color: blue; 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