document.write( "Question 210653: Hi I am having trouble with the following problem.
\n" ); document.write( "Determine whether the following two planes 4x+y-z=3 and 2x-5y+3z=2 are parallel, orthogonal, coincident (that is, the same) or none of these?
\n" ); document.write( "Help with this would be much appreciated.
\n" ); document.write( "Thanks, Judy \n" ); document.write( "
Algebra.Com's Answer #159297 by Alan3354(69443)\"\" \"About 
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Determine whether the following two planes 4x+y-z=3 and 2x-5y+3z=2 are parallel, orthogonal, coincident (that is, the same) or none of these?
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\n" ); document.write( "Find the vectors normal to each plane:
\n" ); document.write( "4x+y-z=3 --> 4i + j - k = normal vector A
\n" ); document.write( "2x-5y+3z=2 --> 2i - 5j + 3k = normal vector B
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\n" ); document.write( "The dot product of the vectors is |A|*|B|cos(a) where a is the angle between the planes.
\n" ); document.write( "The magnitude of A, |A|, is sqrt(16 + 1 + 1) = 2sqrt(3)
\n" ); document.write( "The magnitude of B, |B|, is sqrt(4 + 25 + 9) = sqrt(38)
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\n" ); document.write( "Dot product = 4*2 + 1*(-5) -1*3 = 0
\n" ); document.write( "Since neither A or B is 0, the cos(a) = 0
\n" ); document.write( "a = 90º, they're orthogonal \n" ); document.write( "
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