SOLUTION: Hi I am having trouble with the following problem. Determine whether the following two planes 4x+y-z=3 and 2x-5y+3z=2 are parallel, orthogonal, coincident (that is, the same) or n

Question 210653: Hi I am having trouble with the following problem.
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?
Help with this would be much appreciated.
Thanks, Judy
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
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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Find the vectors normal to each plane:
4x+y-z=3 --> 4i + j - k = normal vector A
2x-5y+3z=2 --> 2i - 5j + 3k = normal vector B
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The dot product of the vectors is |A|*|B|cos(a) where a is the angle between the planes.
The magnitude of A, |A|, is sqrt(16 + 1 + 1) = 2sqrt(3)
The magnitude of B, |B|, is sqrt(4 + 25 + 9) = sqrt(38)
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Dot product = 4*2 + 1*(-5) -1*3 = 0
Since neither A or B is 0, the cos(a) = 0
a = 90�, they're orthogonal
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