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# Prove some equations about distances between points and planes

1. Prove that the distance from the point to the plane

is given by the formula

2. Find the point on the plane which is nearest to the point .

1. Proof. By Theorem 13.6 (page 476 of Apostol) we know that the distance from a point to a plane is given by

2. A normal to the plane is given by . So, for any point . Further, the distance from to a point not on is minimal when where

Thus,

Naming to be the point we have

1. tom says:

b) is likewise incorrect; same answer though (a trick I think)

• Van Gogh says:

For b) I got a different answer… P = (3,1,-5). The given answer 1/25(5,-14,2) does not even seem to be on the plane of 5x-14y+2z=-9.

• Artem says:

Got the same answer: (3, 1, -5). Answer for (b) is incorrect in both Apostol, and above – the found point is not on the plane.

• S says:

I also got (3, 1, -5).

2. tom says:

a) correct statement is distance equals projection of P-Q onto N, not projection P onto N. In Cartesian equation then, d is -N⋅P while N⋅Q is the value of given point.