document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1049882>Question 1049882</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the distance between the lines 2x-2y+3z-12=0=2x+2y+z ,  2x-z=0=5x-2y+9 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #666326 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=666326\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  2x-2y+3z-12 = 0 = 2x+2y+z represent the line that is the intersection of the two planes 2x-2y+3z-12 = 0 and  2x+2y+z = 0.  Similarly,  2x-z = 0 = 5x-2y+9 represent the intersection of the planes 2x-z = 0 and 5x-2y+9 = 0.<BR>\n" );
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document.write( "In the first pair of planes, solving for x and y in terms of z gives x + z = 3 and y - z/2 = -3.  The symmetric equation of the line of intersection is thus <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-3%29%2F%28-1%29+=+%28y%2B3%29%2F%281%2F2%29+=+z ALIGN=MIDDLE  ALT=\"%28x-3%29%2F%28-1%29+=+%28y%2B3%29%2F%281%2F2%29+=+z\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  This line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%5B1%5D ALIGN=MIDDLE  ALT=\"L%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > has direction vector < -1,1/2,1 >, or < -2,1,2 > with the fraction cleared, and passes through the point (3,-3,0).\r<BR>\n" );
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document.write( "The second pair of planes would give a line of intersection with symmetric equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F%281%2F2%29+=+%28y-9%2F2%29%2F%285%2F4%29+=+z ALIGN=MIDDLE  ALT=\"x%2F%281%2F2%29+=+%28y-9%2F2%29%2F%285%2F4%29+=+z\"   DISABLED_style= \"max-width:90%; height:auto;\" >, with direction vector < 1/2, 5/4,1 >, or < 2,5,4 > with the fractions cleared.  This line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%5B2%5D ALIGN=MIDDLE  ALT=\"L%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > line passes through the point (0,9/2,0).\r<BR>\n" );
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document.write( "The vector <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5Bo%5D ALIGN=MIDDLE  ALT=\"v%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > starting at the point (0,9/2,0) of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%5B2%5D ALIGN=MIDDLE  ALT=\"L%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and terminating at the point (3,-3,0) of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%5B1%5D ALIGN=MIDDLE  ALT=\"L%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is given by < 3,-15/2,0 >, or < 6,-15,0 >, with the fraction cleared.\r<BR>\n" );
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document.write( "Now a vector N perpendicular to both the direction vectors of the two intersection lines is < 1,-2,2 >.  (This can be obtained either by getting the cross product of the two direction vectors, or by solving simultaneously the system < a,b,c >*< -2,1,2 > = 0 and < a,b,c >*< 2,5,4 > = 0 for a,b, and c.)\r<BR>\n" );
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document.write( "The distance between the two lines is then just the scalar projection of the vector  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5Bo%5D ALIGN=MIDDLE  ALT=\"v%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > onto the vector N:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28v%5Bo%5D%2AN%29%2Fabs%28N%29+=+%286%2B30%2B0%29%2Fsqrt%281%2B4%2B4%29+=+36%2F3+=+highlight%2812%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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