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 #698522 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=698522\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"4\">Let U and V be two vectors of the form\r<BR>\n" );
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document.write( "U = (a,b)<BR>\n" );
document.write( "V = (c,d)\r<BR>\n" );
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document.write( "where a,b,c,d are scalars and are in the set of real numbers.\r<BR>\n" );
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document.write( "Using those definitions, we can say\r<BR>\n" );
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document.write( "U - 3V = 1*U + (-3)*V<BR>\n" );
document.write( "U - 3V = 1*(a,b) + (-3)*(c,d)<BR>\n" );
document.write( "U - 3V = (a,b) + (-3c,-3d)<BR>\n" );
document.write( "U - 3V = (a-3c,b-3d)\r<BR>\n" );
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document.write( "Since U-3V also equals (1,5), which is given, this means\r<BR>\n" );
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document.write( "(a-3c,b-3d) = (1,5)\r<BR>\n" );
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document.write( "further breaking down to\r<BR>\n" );
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document.write( "a-3c = 1<BR>\n" );
document.write( "b-3d = 5\r<BR>\n" );
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document.write( "Let's refer to these equations as (1) and (2).\r<BR>\n" );
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document.write( "Similarly,\r<BR>\n" );
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document.write( "-U + V = -1*U + 1*V<BR>\n" );
document.write( "-U + V = -1*(a,b) + 1*(c,d)<BR>\n" );
document.write( "-U + V = (-a,-b) + (c,d)<BR>\n" );
document.write( "-U + V = (-a+c,-b+d)\r<BR>\n" );
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document.write( "Since -U + V also equals (-5,3), this means\r<BR>\n" );
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document.write( "(-a+c,-b+d) = (-5,3)\r<BR>\n" );
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document.write( "further breaking down into\r<BR>\n" );
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document.write( "-a+c = -5<BR>\n" );
document.write( "-b+d = 3\r<BR>\n" );
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document.write( "Let's refer to these equations as (3) and (4).\r<BR>\n" );
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document.write( "Group equation (1) and (3) together to get this system\r<BR>\n" );
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document.write( "a-3c = 1<BR>\n" );
document.write( "-a+c = -5\r<BR>\n" );
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document.write( "Adding those equations together leads to -2c = -4 so c = 2\r<BR>\n" );
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document.write( "If c = 2, then...<BR>\n" );
document.write( "a-3c = 1<BR>\n" );
document.write( "a-3(2) = 1<BR>\n" );
document.write( "a-6 = 1<BR>\n" );
document.write( "a-6+6 = 1+6<BR>\n" );
document.write( "a = 7\r<BR>\n" );
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document.write( "So far we know that a = 7 and c = 2\r<BR>\n" );
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document.write( "Now group equation (2) and (4) together and solve the system\r<BR>\n" );
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document.write( "b-3d = 5<BR>\n" );
document.write( "-b+d = 3\r<BR>\n" );
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document.write( "Adding those equations leads to -2d = 8 so d = -4\r<BR>\n" );
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document.write( "If d = -4, then, <BR>\n" );
document.write( "-b+d = 3<BR>\n" );
document.write( "-b+(-4) = 3<BR>\n" );
document.write( "-b-4 = 3<BR>\n" );
document.write( "-b-4+4 = 3+4<BR>\n" );
document.write( "-b = 7<BR>\n" );
document.write( "b = -7\r<BR>\n" );
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document.write( "To wrap up everything so far, we found that\r<BR>\n" );
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document.write( "a = 7<BR>\n" );
document.write( "b = -7<BR>\n" );
document.write( "c = 2<BR>\n" );
document.write( "d = -4\r<BR>\n" );
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document.write( "making <BR>\n" );
document.write( "U = (a,b) and V = (c,d)<BR>\n" );
document.write( "turn into<BR>\n" );
document.write( "U = (7,-7) and V = (2,-4)\r<BR>\n" );
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document.write( "We have the coordinates of U and V, so we can now compute the angle theta between them.\r<BR>\n" );
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document.write( "First we'll need the dot product\r<BR>\n" );
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document.write( "U dot V = a*c + b*d<BR>\n" );
document.write( "U dot V = 7*2 + (-7)*(-4)<BR>\n" );
document.write( "U dot V = 14 + 28<BR>\n" );
document.write( "U dot V = 42\r<BR>\n" );
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document.write( "Now we need the length of each vector\r<BR>\n" );
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document.write( "Let's find the length of vector U\r<BR>\n" );
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document.write( "|U| = sqrt(U dot U)<BR>\n" );
document.write( "|U| = sqrt(a*a + b*b)<BR>\n" );
document.write( "|U| = sqrt(a^2 + b^2)<BR>\n" );
document.write( "|U| = sqrt(7^2 + (-7)^2)<BR>\n" );
document.write( "|U| = sqrt(49 + 49)<BR>\n" );
document.write( "|U| = sqrt(49*2)<BR>\n" );
document.write( "|U| = sqrt(49)*sqrt(2)<BR>\n" );
document.write( "|U| = 7*sqrt(2)\r<BR>\n" );
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document.write( "And we also need the length of vector V\r<BR>\n" );
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document.write( "|V| = sqrt(V dot V)<BR>\n" );
document.write( "|V| = sqrt(c*c + d*d)<BR>\n" );
document.write( "|V| = sqrt(c^2 + d^2)<BR>\n" );
document.write( "|V| = sqrt(2^2 + (-4)^2)<BR>\n" );
document.write( "|V| = sqrt(4 + 16)<BR>\n" );
document.write( "|V| = sqrt(20)<BR>\n" );
document.write( "|V| = sqrt(4*5)<BR>\n" );
document.write( "|V| = sqrt(4)*sqrt(5)<BR>\n" );
document.write( "|V| = 2*sqrt(5)\r<BR>\n" );
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document.write( "It's a lot of work, but we're at the home stretch. Plug the dot product result, and the vector lengths, into the formula below to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta ALIGN=MIDDLE  ALT=\"theta\"   DISABLED_style= \"max-width:90%; height:auto;\" > (greek letter theta) which is the angle between the two vectors U and V\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=theta+=+arccos%28%28dot_product%29%2F%28%28length_of_U%29%2A%28length_of_V%29%29%29 ALIGN=MIDDLE  ALT=\"theta+=+arccos%28%28dot_product%29%2F%28%28length_of_U%29%2A%28length_of_V%29%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+=+arccos%28%2842%29%2F%28%287%2Asqrt%282%29%29%2A%282%2Asqrt%285%29%29%29%29 ALIGN=MIDDLE  ALT=\"theta+=+arccos%28%2842%29%2F%28%287%2Asqrt%282%29%29%2A%282%2Asqrt%285%29%29%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+=+arccos%280.94868329805051%29 ALIGN=MIDDLE  ALT=\"theta+=+arccos%280.94868329805051%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+=+18.4349488229 ALIGN=MIDDLE  ALT=\"theta+=+18.4349488229\"   DISABLED_style= \"max-width:90%; height:auto;\" > Be sure to be in degree mode\r<BR>\n" );
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document.write( "Final Answer: <font color=red>18.4349488229 degrees</font>\r<BR>\n" );
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document.write( "The answer is approximate. Round it however you need to.\r<BR>\n" );
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document.write( "Side Note: The answer has been confirmed with <a  rel=nofollow HREF= \"https://www.geogebra.org/home?ggbLang=en\">GeoGebra</a> (free graphing software) as shown below<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/kYnQrJY.png\"><BR>\n" );
document.write( "The vectors \"check1\" and \"check2\" are defined to be <BR>\n" );
document.write( "check1 = U - 3V<BR>\n" );
document.write( "check2 = -U + V<BR>\n" );
document.write( "so that part is confirmed as well.</font> </SPAN>\n" );
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