- Prove that the distance from the point to the plane
is given by the formula
- Find the point on the plane which is nearest to the point .
- Proof. By Theorem 13.6 (page 476 of Apostol) we know that the distance from a point to a plane is given by
- 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
b) is likewise incorrect; same answer though (a trick I think)
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.
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.
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.