document.write( "Question 1158864: Find the equation of the plain passing through the points P(2, -3, 1), P’(5, -3, -5) and perpendicular to the plane x - 2y + 5z + 20 = 0.\r
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Algebra.Com's Answer #781959 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Normal vectors to the two planes must have dot product 0 in order\r\n" );
document.write( "for them to be perpendicular.\r\n" );
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document.write( "The coefficients of x, y and z in the equation of a plane \r\n" );
document.write( "are the components of a vector normal to that plane.\r\n" );
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document.write( "Let the desired plane have an equation\r\n" );
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document.write( "\"ax%2Bby%2Bcz%2Bd=0\"\r\n" );
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document.write( "then it will have a normal vector < a, b, c >\r\n" );
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document.write( "The two given points must satisfy the equation of the\r\n" );
document.write( "desired plane.  So we have these two equations:\r\n" );
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document.write( "a(2)+b(-3)+c(1)+d=0\r\n" );
document.write( "a(5)+b(-3)+c(-5)+d=0\r\n" );
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document.write( "which simplify to\r\n" );
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document.write( "2a-3b+ c+d=0\r\n" );
document.write( "5a-3b-5c+d=0\r\n" );
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document.write( "The coefficients of x, y and z of the plane  x - 2y + 5z + 20 = 0\r\n" );
document.write( "are the components of a normal vector to that plane.  And the dot\r\n" );
document.write( "product of vectors normal to them must be 0.\r\n" );
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document.write( "< 1, -2, 5 > • < a, b, c > = 0\r\n" );
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document.write( "1a - 2b + 5c = 0\r\n" );
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document.write( "So we have the system of \r\n" );
document.write( "equations:\r\n" );
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document.write( "\"system%282a-3b%2B+c%2Bd=0%2C%0D%0A5a-3b-5c%2Bd=0%2Ca+-+2b+%2B+5c+=+0%29\"\r\n" );
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document.write( "Solve that by the Gauss-Jordan method\r\n" );
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document.write( "Use your TI-84 to get the rref of that matrix, which is\r\n" );
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document.write( "Which tells us\r\n" );
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document.write( "\"system%28a-expr%284%2F11%29d=0%2Cb-expr%287%2F11%29d=0%2C+c-expr%282%2F11%29d=0%29\"\r\n" );
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document.write( "or\r\n" );
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document.write( "\"system%28a=expr%284%2F11%29%2Ad%2Cb=expr%287%2F11%29%2Ad%2C+c=expr%282%2F11%29%2Ad%29\"\r\n" );
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document.write( "So one equation for the desired plane could be\r\n" );
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document.write( "\"expr%284%2F11%29%2Ad%2Ax%2Bexpr%287%2F11%29%2Ad%2Ay%2Bexpr%282%2F11%29%2Ad%2Az%2Bd=0\"\r\n" );
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document.write( "We may multiply through by 11\r\n" );
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document.write( "\"4d%2Ax%2B7d%2Ay%2B2d%2Az%2B11%2Ad=0\"\r\n" );
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document.write( "And then divide through by d\r\n" );
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document.write( "\"4x%2B7y%2B2z%2B11=0\"   \r\n" );
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document.write( "That's the best answer.\r\n" );
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document.write( "Edwin
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