document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1109182>Question 1109182</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The points A, B, C and D have position vectors i +j +k, 2i+3j, 3i+5j-2k and -j+k respectively. find which of the following pairs of lines are parallel. a) AB and CD   b) BC and CD   c) BC and AD </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #724247 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=724247\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Looking at the vector addition,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2BAB=B ALIGN=MIDDLE  ALT=\"A%2BAB=B\"   DISABLED_style= \"max-width:90%; height:auto;\" > so <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=B-A ALIGN=MIDDLE  ALT=\"AB=B-A\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Similarly,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=BC=C-B ALIGN=MIDDLE  ALT=\"BC=C-B\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CD=D-C ALIGN=MIDDLE  ALT=\"CD=D-C\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AD=D-A ALIGN=MIDDLE  ALT=\"AD=D-A\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "A:(1,1,1)<BR>\n" );
document.write( "B:(2,3,0)<BR>\n" );
document.write( "C:(3,5,-2)<BR>\n" );
document.write( "D:(0,-1,1)<BR>\n" );
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document.write( "AB=B-A=(2,3,0)-(1,1,1)=(2-1,3-1,0-1)=(1,2,-1)<BR>\n" );
document.write( "BC=C-B=(3,5,-2)-(2,3,0)=(3-2,5-3,-2-0)=(1,2,-2)<BR>\n" );
document.write( "CD=D-C=(0,-1,1)-(3,5,-2)=(0-3,-1-5,1-(-2))=(-3,6,3)<BR>\n" );
document.write( "AD=D-A=(0,-1,1)-(1,1,1)=(0-1,-1-1,1-1)=(-1,-2,0)<BR>\n" );
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document.write( "You can either find the unit vectors of each vector and compare or you can look for vectors that are multiples of each other, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Y=kX ALIGN=MIDDLE  ALT=\"Y=kX\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > is an integer. <BR>\n" );
document.write( "Upon inspection you can see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=CD=-3%2AAB ALIGN=MIDDLE  ALT=\"CD=-3%2AAB\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so CD and AB are parallel.<BR>\n" );
document.write( " </SPAN>\n" );
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