document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=204902>Question 204902</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi all, Im still having trouble with these vector problems, I cant solve the problems listed below for the following: <BR>\n" );
document.write( "Let ~u = 3~i - ~j - 2~k and ~v = 2~i + ~j + ~k.(Again, the ~ symbols mean that there is and arrow symbol above the value in the positive direction)<BR>\n" );
document.write( "I need to:<BR>\n" );
document.write( "a) Calculate the cosine of the angle between ~u and ~v.<BR>\n" );
document.write( "b) Fin the cross product of ~u and ~v.<BR>\n" );
document.write( "Some help with steps and explanations on how to solve would be great.<BR>\n" );
document.write( "Thanks, -Nick </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #154652 by </B><A HREF=/tutors/aboutme.mpl?userid=alicealc><B>alicealc(293)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=alicealc><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=154652\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> u = 3i - j - 2k<BR>\n" );
document.write( "v = 2i + j + k\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "size of u (|u|) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%5E2%2B%28-1%29%5E2%2B%28-2%29%5E2%29+=+sqrt%289%2B1%2B4%29=sqrt%2814%29 ALIGN=MIDDLE  ALT=\"sqrt%283%5E2%2B%28-1%29%5E2%2B%28-2%29%5E2%29+=+sqrt%289%2B1%2B4%29=sqrt%2814%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "size of v (|v|) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%5E2%2B1%5E2%2B1%5E2%29+=+sqrt%284%2B1%2B1%29=sqrt%286%29 ALIGN=MIDDLE  ALT=\"sqrt%282%5E2%2B1%5E2%2B1%5E2%29+=+sqrt%284%2B1%2B1%29=sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "a) cosine of the angle between u and v:<BR>\n" );
document.write( "cos(angle) = (dot product between u and v)/(|u|*|v|)<BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2A2%2B%28-1%29%2A1%2B%28-2%29%2A1%29%2F%28sqrt%2814%29%2Asqrt%286%29%29 ALIGN=MIDDLE  ALT=\"%283%2A2%2B%28-1%29%2A1%2B%28-2%29%2A1%29%2F%28sqrt%2814%29%2Asqrt%286%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286-1-2%29%2Fsqrt%2884%29 ALIGN=MIDDLE  ALT=\"%286-1-2%29%2Fsqrt%2884%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Fsqrt%284%2A21%29 ALIGN=MIDDLE  ALT=\"3%2Fsqrt%284%2A21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F%282%2Asqrt%2821%29%29 ALIGN=MIDDLE  ALT=\"3%2F%282%2Asqrt%2821%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "it can be rationalized so it becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F%282%2Asqrt%2821%29%29%29%2A%28sqrt%2821%29%2Fsqrt%2821%29%29 ALIGN=MIDDLE  ALT=\"%283%2F%282%2Asqrt%2821%29%29%29%2A%28sqrt%2821%29%2Fsqrt%2821%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Asqrt%2821%29%2F%282%2A21%29 ALIGN=MIDDLE  ALT=\"3%2Asqrt%2821%29%2F%282%2A21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2821%29%2F%282%2A7%29 ALIGN=MIDDLE  ALT=\"sqrt%2821%29%2F%282%2A7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2821%29%2F14 ALIGN=MIDDLE  ALT=\"sqrt%2821%29%2F14\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "b) i cross j = k, j cross k = i, k cross i = j<BR>\n" );
document.write( "j cross i = -k, k cross j = -i, i cross k = -j<BR>\n" );
document.write( "cross product between i and i = between j and j = between k and k = 0<BR>\n" );
document.write( "so,cross product between u and v:<BR>\n" );
document.write( "(3i*j) + (3i*k) + (-j*2i) + (-j*k) + (-2k*2i) + (-2k*j)<BR>\n" );
document.write( "= 3k + (-3j) + 2k + (-i) +(-4j) + 2i<BR>\n" );
document.write( "= i - 7j + 5k </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );