document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=206224>Question 206224</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi again, hopefully someone can help me with the follwoing question,<BR>\n" );
document.write( "Find an equation of the line intersection of the two planes:<BR>\n" );
document.write( "x+2y-z = -1 and 3x-2y+5z = 5<BR>\n" );
document.write( "Any help would be most appreciated.<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 #155816 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=155816\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find an equation of the line intersection of the two planes:<BR>\n" );
document.write( "x+2y-z = -1 and 3x-2y+5z = 5<BR>\n" );
document.write( "---------------<BR>\n" );
document.write( "The vector v1 = (1,2,-1) is normal to x+2y-z = -1.<BR>\n" );
document.write( "The vector v2 = (3,-2,5) is normal to 3x-2y+5z = 5.<BR>\n" );
document.write( "These vectors are not parallel, so the planes do intersect.<BR>\n" );
document.write( "-----------------<BR>\n" );
document.write( "The cross product gives a vector that's perpendicular to both above.<BR>\n" );
document.write( "v3 = v1Xv2 = i - j - k (Cross product of v1 and v2), or (1,-1,-1)<BR>\n" );
document.write( "v3 gives the direction of the line of intersection.<BR>\n" );
document.write( "--------------<BR>\n" );
document.write( "Next, we need a point that is on both planes.<BR>\n" );
document.write( "+x+2y-z = -1<BR>\n" );
document.write( "3x-2y+5z = 5<BR>\n" );
document.write( "--------------<BR>\n" );
document.write( "Set one dimension to zero, and get 2 linear eqns:<BR>\n" );
document.write( "x=0<BR>\n" );
document.write( "2y - z = -1<BR>\n" );
document.write( "2y -5z = -5<BR>\n" );
document.write( "----------- Subtract 2 from 1<BR>\n" );
document.write( "4z = 4<BR>\n" );
document.write( "z = 1<BR>\n" );
document.write( "y = 0<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "So the point with the position vector (0,0,1) lies on the line of intersection.<BR>\n" );
document.write( "The eqn of the line of intersection is (0,0,1) + t(1,-1,-1)<BR>\n" );
document.write( "-----------------<BR>\n" );
document.write( "To check a few points:<BR>\n" );
document.write( "+x+2y-z = -1<BR>\n" );
document.write( "3x-2y+5z = 5<BR>\n" );
document.write( "t = 1 --> (1,-1,0) is on both planes.<BR>\n" );
document.write( "t = 4 --> (4,-4,-3) is also on both planes.<BR>\n" );
document.write( "2 points is sufficient to fix a line, so it's correct.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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