document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202786>Question 1202786</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Has this problem been resolved correctly?\r<BR>\n" );
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document.write( "Find the angle between two vectors u=3i+4j and v=5i-4j<BR>\n" );
document.write( "        3 + 4 and 5 − 4<BR>\n" );
document.write( "        3 * 5 + 4 * −4<BR>\n" );
document.write( "          15  +   −16<BR>\n" );
document.write( "             -1\r<BR>\n" );
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document.write( "||u|| = √(3^2+4²) = √(9+16) = √25 = √5<BR>\n" );
document.write( "||v|| = √(5^2+(-4)²) = √(25+16) = √41\r<BR>\n" );
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document.write( "θ = cos^1(-1/√205) = cos^1 (-(1/√205*√205/√205)) <BR>\n" );
document.write( "= cos⁡-√205/√205 = cos⁡√205/√205\r<BR>\n" );
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document.write( "Thank you<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #837882 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=837882\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "You have the correct dot product of -1.\r<BR>\n" );
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document.write( "The computation for ||u|| is incorrect. It should be 5 instead of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29 ALIGN=MIDDLE  ALT=\"sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The computation for ||v|| is correct. \r<BR>\n" );
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document.write( "cos(theta) = ( u dot v )/( ||u||*||v|| )<BR>\n" );
document.write( "cos(theta) = ( -1 )/( 5*sqrt(41) )<BR>\n" );
document.write( "cos(theta) = -0.03123475237772<BR>\n" );
document.write( "theta = arccos(-0.03123475237772)<BR>\n" );
document.write( "theta = 91.789910608246 degrees\r<BR>\n" );
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document.write( "The result is approximate. Round it however needed.<BR>\n" );
document.write( "The answer can be confirmed with GeoGebra.<BR>\n" );
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