document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=91469>Question 91469</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Approximate the angle between the two vectors 4i-3j and 4i+3j.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #66426 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=66426\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To calculate the angle between the vectors <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE <4,-3>\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE <4,3>\"> 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 \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( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \cos(\theta)=\frac{4*4+-3*3}{sqrt{4*4+-3*-3}sqrt{4*4+3*3}}=\frac{16+-9}{sqrt{16+9}sqrt{16+9}}=\frac{7}{sqrt{25}sqrt{25}}=\frac{7}{sqrt{625}}\">Calculate the dot product in the numerator and the magnitudes in the denominator\r<BR>\n" );
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document.write( "So 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{7}{25}\">\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{7}{25}\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{7}{25}\right)=\cos^{-1}\left(0.28\left)=1.28700221758657\">\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=1.28700221758657\"> radians\r<BR>\n" );
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document.write( "The angle in degrees is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=(1.28700221758657)*\frac{180}{\pi}=73.739795291688\">\r<BR>\n" );
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document.write( "So the angle is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \theta=73.739795291688\"> degrees\r<BR>\n" );
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