document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1084407>Question 1084407</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If I've:   (U-3V)=(1;5) and (-U+V)=(-5;3)\r<BR>\n" );
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document.write( "What's the angle between U and V?? Thank you VERY MUCH !!  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #698523 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=698523\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1)  (U-3V) = (1;5)<BR>\n" );
document.write( "2)  (-U+V) = (-5;3)<BR>\n" );
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document.write( "solve equation 1 for U<BR>\n" );
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document.write( "U = (1;5) + 3V<BR>\n" );
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document.write( "substitute in equation 2<BR>\n" );
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document.write( "-[(1;5) + 3V] + V= (-5;3)<BR>\n" );
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document.write( "(-1;-5) -3V + V = (-5;3)<BR>\n" );
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document.write( "-2V = (-4;8)<BR>\n" );
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document.write( "V = (2;-4)<BR>\n" );
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document.write( "U = (1;5) + 3(2;-4) = (7;-7)<BR>\n" );
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document.write( "U is (7,-7) and V is (2,-4)<BR>\n" );
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document.write( "we want the angle(theta) between vectors U and V<BR>\n" );
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document.write( "cos theta = (U . V) / |U| |V|, where . means dot product<BR>\n" );
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document.write( "U . V = 7*2 + -7*-4 = 42<BR>\n" );
document.write( "|U| = square root(49 + 49) = square root(98) = 7 * square root(2)<BR>\n" );
document.write( "|V| = square root(4 + 16) = square root(20) = 2 * square root(5)<BR>\n" );
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document.write( "cos theta = 42 / 14 * square root(2) * square root(5) = 3 / square root(10) = 0.9487<BR>\n" );
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document.write( "inverse cos 0.9487 = 18.4 degrees<BR>\n" );
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document.write( "The angle between vectors U and V is 18.4 degrees<BR>\n" );
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