document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=764696>Question 764696</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the area of a triangle wit vertices at ( 3, -1, 2 ) ( 1,-1,-3) and (( 4,-3,-1) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #465664 by </B><A HREF=/tutors/aboutme.mpl?userid=Lola77479><B>Lola77479(27)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Lola77479><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=465664\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> find the area of a triangle wit vertices at ( 3, -1, 2 ) ( 1,-1,-3) and (( 4,-3,-1)<BR>\n" );
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document.write( "Find the length of each side.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+sqrt%28diffx%5E2+%2B+diffy%5E2+%2B+diffz%5E2%29 ALIGN=MIDDLE  ALT=\"s+=+sqrt%28diffx%5E2+%2B+diffy%5E2+%2B+diffz%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then use Heron's Law to find the area.<BR>\n" );
document.write( "I looked at that, it's possible but tedious.<BR>\n" );
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document.write( "Another approach:<BR>\n" );
document.write( "Find the lengths of 2 sides, a & b, and the angle between them.<BR>\n" );
document.write( "Area = (a*b*sine(angle))/2<BR>\n" );
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document.write( "Label the point.<BR>\n" );
document.write( "A(3,-1,2) B(1,-1,-3) and C(4,-3,-1)<BR>\n" );
document.write( "Find 2 vectors for 2 of the sides<BR>\n" );
document.write( "U = AB = (2,0,5)<BR>\n" );
document.write( "V = AC = (-1,2,3)<BR>\n" );
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document.write( "Get UxV the cross product<BR>\n" );
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document.write( "|+i +j +k|<BR>\n" );
document.write( "|+2 +0 +5|<BR>\n" );
document.write( "|-1 +2 +3|<BR>\n" );
document.write( "--> i*(0-10) - j*(6+5) + k*(4 +0)<BR>\n" );
document.write( "UxV = -10i - 11j + 4k<BR>\n" );
document.write( "Find the magnitude of UxV<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M+=+sqrt%2810%5E2+%2B+11%5E2+%2B+4%5E2%29+=+sqrt%28227%29 ALIGN=MIDDLE  ALT=\"M+=+sqrt%2810%5E2+%2B+11%5E2+%2B+4%5E2%29+=+sqrt%28227%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Area+=+sqrt%28227%29%2F2 ALIGN=MIDDLE  ALT=\"Area+=+sqrt%28227%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
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