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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_303</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_302.html">< previous page</a></td> <td id="ebook_previous" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_303</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_305.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 303</font></td></tr></table><table border="0" cellspacing="0" cellpadding="0"><tr><td rowspan="5"></td> <td colspan="3" height="17"></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">B.7<br />Miscellaneous Comments</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">Let R</font><font face="Times New Roman, Times, Serif" size="1"><sub><i>a</i>2<i>b</i></sub></font><font face="Times New Roman, Times, Serif" size="3"> represent the coordinate transformation from reference frame <i>a</i> to reference frame <i>b</i>. The inverse transformation can be shown to be <img src="4e688f755319cf50792bc56a9adb7075.gif" border="0" alt="C0303-01.GIF" width="33" height="25" />. Then <img src="e19d89068517441709077136598206da.gif" border="0" alt="C0303-02.GIF" width="181" height="24" />. This property implies that R</font><font face="Times New Roman, Times, Serif" size="1"><sub><i>a</i>2<i>b</i></sub></font><font face="Times New Roman, Times, Serif" size="3"> is an orthogonal matrix.</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">The following are formal definitions for scalars and vectors [105]:</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">Scalars are quantities that are invariant under coordinate transformation (e.g., mass).</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">Vectors are sets of quantities that, if organized as v = [</font><font face="Symbol" size="3"><i>u</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>1</sub></font><font face="Times New Roman, Times, Serif" size="3">, </font><font face="Symbol" size="3"><i>u</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>2</sub></font><font face="Times New Roman, Times, Serif" size="3">, </font><font face="Symbol" size="3"><i>u</i></font><font face="Times New Roman, Times, Serif" size="1"><sub>3</sub></font><font face="Times New Roman, Times, Serif" size="3">]</font><font face="Times New Roman, Times, Serif" size="2"><sup><i>T</i></sup></font><font face="Times New Roman, Times, Serif" size="3">, transform between coordinate frames according to (v)</font><font face="Times New Roman, Times, Serif" size="2"><sup><i>b</i></sup></font><font face="Times New Roman, Times, Serif" size="3"> = R</font><font face="Times New Roman, Times, Serif" size="1"><sub><i>a</i>2<i>b</i></sub></font><font face="Times New Roman, Times, Serif" size="3">(v)</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">.</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">This strict definition of vectors is required for distinguishing which items arranged as [<i>a, b, c</i>] can be transformed between coordinate systems by means of vector transformation operations (i.e., rotations). Some quantities [e.g., the finite rotations (</font><font face="Symbol" size="3"><i>f</i></font><i><font face="Times New Roman, Times, Serif" size="3">, </font><font face="Symbol" size="3">q</font><font face="Times New Roman, Times, Serif" size="3">, </font><font face="Symbol" size="3">y</font></i><font face="Symbol" size="3"></font><font face="Times New Roman, Times, Serif" size="3">)] although convenient to organize in a vectorlike notation, are not vectors in the strict sense defined above. The Euler angle three-tuple (</font><font face="Symbol" size="3"><i>f</i></font><i><font face="Times New Roman, Times, Serif" size="3">, </font><font face="Symbol" size="3">q</font><font face="Times New Roman, Times, Serif" size="3">, </font><font face="Symbol" size="3">y</font></i><font face="Symbol" size="3"></font><font face="Times New Roman, Times, Serif" size="3">) is not a vector. The final orientation associated with this three-tuple is dependent on an assumed order of rotation.</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></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; font-size: 120%; font-weight: bold; text-decoration: none; font-family: verdana;" href="page_302.html">< previous page</a></td> <td id="ebook_next" align="center" width="40%" style="background: #EEF3E2"><strong style="color: #2F4F4F; font-size: 120%;">page_303</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_305.html">next page ></a></td> </tr> </table> </body> </html>⌨️ 快捷键说明
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