document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1048763>Question 1048763</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the general form of the equation of the plane with the given characteristics.<BR>\n" );
document.write( "The plane passes through the points, (-4,-7,7) and (-2,-1,-6)<BR>\n" );
document.write( "and is perpendicular to the plane x-3y+5z=-1.<BR>\n" );
document.write( "*Please explain in detail!!!* </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #664349 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=664349\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A normal vector to the plane x-3y+5z=-1 is < 1,-3, 5 >.\r<BR>\n" );
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document.write( "Since the points (-4,-7,7) and (-2,-1,-6) are on the desired plane, a vector parallel to the desired plane is < -2--4, -1--7, -6-7 > = < 2,6, -13 >.\r<BR>\n" );
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document.write( "Let < a,b,c > be the normal vector of the desired plane.\r<BR>\n" );
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document.write( "< 2,6, -13 >*< a,b,c > = 0, or \r<BR>\n" );
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document.write( "2a + 6b - 13c = 0  <------Equation A\r<BR>\n" );
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document.write( "Also,  < 1,-3,5 >*< a,b,c > = 0, or \r<BR>\n" );
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document.write( "a - 3b + 5c = 0,  <-------Equation B\r<BR>\n" );
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document.write( "as per the given that the two planes are perpendicular.\r<BR>\n" );
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document.write( "Multiplying Equation B by 2 and adding with Equation A gives \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+%283%2F4%29c ALIGN=MIDDLE  ALT=\"a+=+%283%2F4%29c\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Multiplying Equation B by -2 and adding with Equation A gives \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+%2823%2F12%29c ALIGN=MIDDLE  ALT=\"b+=+%2823%2F12%29c\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "===> The normal vector < a,b,c > = c < 3/4, 23/12,1 >.\r<BR>\n" );
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document.write( "Choosing c = 12, let the normal be < 9, 23, 12 >.\r<BR>\n" );
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document.write( "Therefore, the desired plane has the equation\r<BR>\n" );
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document.write( "9(x+2)+23(y+1)+12(z+6) = 0.\r<BR>\n" );
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document.write( "I leave the simplifying to you.<BR>\n" );
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