document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=334795>Question 334795</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How can I use the vector product to find the angle between the vectors:<BR>\n" );
document.write( "v= 6i - 3j + k and w= 2i + 2j - k. ???<BR>\n" );
document.write( "Any help would be great.<BR>\n" );
document.write( "-Nick </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #239910 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=239910\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> How can I use the vector product to find the angle between the vectors:<BR>\n" );
document.write( "v= 6i - 3j + k and w= 2i + 2j - k.<BR>\n" );
document.write( "------------<BR>\n" );
document.write( "The dot product is the product of the 2 magnitudes * cos of the angle between the vectors.<BR>\n" );
document.write( "|v| = sqrt(36 + 9 + 1) = sqrt(46)<BR>\n" );
document.write( "|w| = sqrt(4 + 4 + 1) = 3<BR>\n" );
document.write( "----------------<BR>\n" );
document.write( "v dot w = 6*2 + -3*2 + -1 = 5<BR>\n" );
document.write( "5 = 3*sqrt(46)*cos(A)<BR>\n" );
document.write( "cos(A) = 5/(3sqrt(46)) =~ 0.2457366<BR>\n" );
document.write( "Angle =~ 75.7746 degs<BR>\n" );
document.write( " <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );