Section 12.3 : Equations of Planes
4. Determine if the plane given by 4x−9y−z=2 and the plane given by x+2y−14z=−6 are parallel, orthogonal or neither.
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Start SolutionLet’s start off this problem by noticing that the vector →n1=⟨4,−9,−1⟩ will be normal to the first plane and the vector →n2=⟨1,2,−14⟩ will be normal to the second plane.
Now try to visualize the two planes and these normal vectors. What would the two planes look like if the two normal vectors were parallel to each other? What would the two planes look like if the two normal vectors were orthogonal to each other?
Show Step 2If the two normal vectors are parallel to each other the two planes would also need to be parallel.
So, let’s take a quick look at the normal vectors. We can see that the first component of each vector have the same sign and the same can be said for the third component. However, the second component of each vector has opposite signs.
Therefore, there is no number that we can multiply to →n1 that will keep the sign on the first and third component the same and simultaneously changing the sign on the second component. This in turn means the two vectors can’t possibly be scalar multiples and this further means they cannot be parallel.
If the two normal vectors can’t be parallel then the two planes are not parallel.
Show Step 3Now, if the two normal vectors are orthogonal the two planes will also be orthogonal.
So, a quick dot product of the two normal vectors gives,
→n1⋅→n2=0The dot product is zero and so the two normal vectors are orthogonal. Therefore, the two planes are orthogonal.