document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1083103>Question 1083103</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (b)	Show that the vectors P = 3i – 2j + k, Q = i – 3j + 5k, and R = 2i + j – 4k form a right angled triangle.<BR>\n" );
document.write( "(c)	Use vectors in (b) above to evaluate P Q and R  Q what do these two final vectors represent<BR>\n" );
document.write( ".thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #698042 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=698042\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The sum of two vectors must equal the other vector,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ%2BQR=PR ALIGN=MIDDLE  ALT=\"PQ%2BQR=PR\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ=OP-OQ ALIGN=MIDDLE  ALT=\"PQ=OP-OQ\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=QR=OQ-OR ALIGN=MIDDLE  ALT=\"QR=OQ-OR\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PR=OP-OR ALIGN=MIDDLE  ALT=\"PR=OP-OR\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where O stands for the origin,<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ=+%28matrix%281%2C5%2C3%2C%22%2C%22%2C-2%2C%22%2C%22%2C1%29%29%2B%28matrix%281%2C5%2C1%2C%22%2C%22%2C-3%2C%22%2C%22%2C5%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ=+%28matrix%281%2C5%2C2%2C%22%2C%22%2C1%2C%22%2C%22%2C-4%29%29 ALIGN=MIDDLE  ALT=\"PQ=+%28matrix%281%2C5%2C2%2C%22%2C%22%2C1%2C%22%2C%22%2C-4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=QR=+%28matrix%281%2C5%2C1%2C%22%2C%22%2C-3%2C%22%2C%22%2C5%29%29%2B%28matrix%281%2C5%2C2%2C%22%2C%22%2C1%2C%22%2C%22%2C4%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=QR=+%28matrix%281%2C5%2C-1%2C%22%2C%22%2C-4%2C%22%2C%22%2C9%29%29 ALIGN=MIDDLE  ALT=\"QR=+%28matrix%281%2C5%2C-1%2C%22%2C%22%2C-4%2C%22%2C%22%2C9%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PR=+%28matrix%281%2C5%2C3%2C%22%2C%22%2C-2%2C%22%2C%22%2C1%29%29%2B%28matrix%281%2C5%2C2%2C%22%2C%22%2C1%2C%22%2C%22%2C4%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PR=+%28matrix%281%2C5%2C1%2C%22%2C%22%2C-3%2C%22%2C%22%2C5%29%29 ALIGN=MIDDLE  ALT=\"PR=+%28matrix%281%2C5%2C1%2C%22%2C%22%2C-3%2C%22%2C%22%2C5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
document.write( "And,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ%2BQR=%28matrix%281%2C5%2C2%2C%22%2C%22%2C1%2C%22%2C%22%2C-4%29%29%2B%28matrix%281%2C5%2C-1%2C%22%2C%22%2C-4%2C%22%2C%22%2C9%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PQ%2BQR=%28matrix%281%2C5%2C1%2C%22%2C%22%2C-3%2C%22%2C%22%2C5%29%29 ALIGN=MIDDLE  ALT=\"PQ%2BQR=%28matrix%281%2C5%2C1%2C%22%2C%22%2C-3%2C%22%2C%22%2C5%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=PQ%2BQR=PR ALIGN=MIDDLE  ALT=\"PQ%2BQR=PR\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the two vectors do add up to the third.<BR>\n" );
document.write( "Additionally, two of the vectors must be perpendicular to each other in order for that to happen.<BR>\n" );
document.write( "So check the dot products,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2AQ=3%281%29%2B%28-2%29%28-3%29%2B1%285%29=3%2B6%2B5=14 ALIGN=MIDDLE  ALT=\"P%2AQ=3%281%29%2B%28-2%29%28-3%29%2B1%285%29=3%2B6%2B5=14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Q%2AR=1%282%29%2B%28-3%29%281%29%2B5%28-4%29=2-3-20=-21 ALIGN=MIDDLE  ALT=\"Q%2AR=1%282%29%2B%28-3%29%281%29%2B5%28-4%29=2-3-20=-21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2AR=3%282%29%2B%28-2%29%281%29%2B1%28-4%29=6-2-4=0 ALIGN=MIDDLE  ALT=\"P%2AR=3%282%29%2B%28-2%29%281%29%2B1%28-4%29=6-2-4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since the dot product is zero, the two vectors are perpendicular (right angle) and together with the result above it proves that the vectors form a right triangle.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );