document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197954>Question 1197954</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the angle Θ between the two vectors u=<3,4> and v<2,1> </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #831449 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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/cgi-bin/show-face.mpl?user=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=831449\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "What is the angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > between the two vectors <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u ALIGN=MIDDLE  ALT=\"u\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >> and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v ALIGN=MIDDLE  ALT=\"v\"   DISABLED_style= \"max-width:90%; height:auto;\" ><<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >>\r<BR>\n" );
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document.write( "The angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > between two vectors  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+ ALIGN=MIDDLE  ALT=\"u+\"   DISABLED_style= \"max-width:90%; height:auto;\" >and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v ALIGN=MIDDLE  ALT=\"v\"   DISABLED_style= \"max-width:90%; height:auto;\" > is related to the modulus (or magnitude) and scaler (or dot) product of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+ ALIGN=MIDDLE  ALT=\"u+\"   DISABLED_style= \"max-width:90%; height:auto;\" >and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v ALIGN=MIDDLE  ALT=\"v\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the relationship:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%2Av=+abs%28u%29%2Aabs%28v%29%2Acos%28theta%29 ALIGN=MIDDLE  ALT=\"u%2Av=+abs%28u%29%2Aabs%28v%29%2Acos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "first find dot product:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%2Av ALIGN=MIDDLE  ALT=\"u%2Av\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >>*<<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%2Av=3%2A2+%2B4%2A1 ALIGN=MIDDLE  ALT=\"u%2Av=3%2A2+%2B4%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%2Av=10 ALIGN=MIDDLE  ALT=\"u%2Av=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "now find the modulus: \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28u%29=sqrt%283%5E2%2B4%5E2%29=sqrt%2825%29=5 ALIGN=MIDDLE  ALT=\"abs%28u%29=sqrt%283%5E2%2B4%5E2%29=sqrt%2825%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28v%29=sqrt%282%5E2%2B1%5E2%29=sqrt%285%29 ALIGN=MIDDLE  ALT=\"abs%28v%29=sqrt%282%5E2%2B1%5E2%29=sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "their product:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28u%29%2Aabs%28v%29=5sqrt%285%29 ALIGN=MIDDLE  ALT=\"abs%28u%29%2Aabs%28v%29=5sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "plug all in formula above:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10=+5sqrt%285%29%2Acos%28theta%29 ALIGN=MIDDLE  ALT=\"10=+5sqrt%285%29%2Acos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2F%285sqrt%285%29%29=cos%28theta%29 ALIGN=MIDDLE  ALT=\"10%2F%285sqrt%285%29%29=cos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Fsqrt%285%29=cos%28theta%29 ALIGN=MIDDLE  ALT=\"2%2Fsqrt%285%29=cos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=cos%5E-1%282%2Fsqrt%285%29%29 ALIGN=MIDDLE  ALT=\"theta=cos%5E-1%282%2Fsqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=0.463647609 ALIGN=MIDDLE  ALT=\"theta=0.463647609\"   DISABLED_style= \"max-width:90%; height:auto;\" >-> radians\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta=26.57 ALIGN=MIDDLE  ALT=\"theta=26.57\"   DISABLED_style= \"max-width:90%; height:auto;\" >°\r<BR>\n" );
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document.write( " </SPAN>\n" );
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