document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=191763>Question 191763</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/1602773033/algebrahomeworkh>saxon algebra2</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Find the angle between the vectors<BR>\n" );
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document.write( "theta arc cos (-13over square root 225 square root 64)<BR>\n" );
document.write( "I don't know what to do to finish this off. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #143876 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=143876\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To calculate the angle <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta\"> between the vectors <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE <-5,3>\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE <2,4>\"> use this formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos(\theta)=\frac{\v{a}\cdot\v{b}}{\left|\v{a}\right|\left|\v{b}\right|}\"> where   <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \v{a}\cdot\v{b}\"> is the dot product of vectors \"a\" and \"b\", <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left|\v{a}\right|\"> is the magnitude (ie length) of vector <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \v{a}\">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left|\v{b}\right|\"> is the magnitude (ie length) of vector <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \v{b}\">\r<BR>\n" );
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document.write( "Note: I'm assuming you are familiar with these concepts.\r<BR>\n" );
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document.write( "So...\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos(\theta)=\frac{-5*2+3*4}{sqrt{-5*-5+3*3}sqrt{2*2+4*4}}=\frac{-10+12}{sqrt{25+9}sqrt{4+16}}=\frac{2}{sqrt{34}sqrt{20}}=\frac{2}{sqrt{680}}\">\r<BR>\n" );
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document.write( "Now we have this equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos(\theta)=\frac{2}{2sqrt(170)}\">\r<BR>\n" );
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document.write( "Now take the arccosine of both sides to isolate <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \theta\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos^{-1}\left(\cos(\theta)\right)=\cos^{-1}\left(\frac{2}{2sqrt(170)}\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=\cos^{-1}\left(\frac{2}{2sqrt(170)}\right)=\cos^{-1}\left(0.076696498884737\left)=1.49402443552512\">\r<BR>\n" );
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document.write( "So\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta \approx 1.49402443552512\"> radians\r<BR>\n" );
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document.write( "Note: The angle in degrees is approximately:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=(1.49402443552512)*\frac{180}{\pi}=85.6012946450045\">\r<BR>\n" );
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document.write( "Or simply \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta \approx 85.6012946450045\"> degrees\r<BR>\n" );
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