Question

Multivariable Calculus:

Find the distance from the point P(-1, 0, 2) to the plane passing through the points A(-2, 1, 1), B(0, 5, 2), and C(1, 3, 0).

Answer 1

there are a few appraoches we can take

Answer 2

I suspect we could do projection to the normal vector.

Answer 3

one approach, is to find a point in the plane that is "in line" with when given point along the normal, and then determine the distance between them.

another amounts to finding the normal and a vector to one of the planar points, and working a cos of the angle between them ...

Answer 4

project onto the normal is fair as well

Answer 5

for the plane passing through A,B,C, look at AB and AC and take their cross product for the plane normal. you can then project the vector AP onto the normal to find the distance

Answer 6

|n| |v| cos(a) = n.v

|n| cos(a) = n.v/|v| is the length of the normal between the point and the plane