Projection of vector onto the plane.

Question

I have vector $b = (1, 1, 1)$ and the plane $2x - y + z = 1$.

I know how to find the projection of the vector on the vector, but how to do this with the plane?

Answer 1

A plane is uniquely defined by a point and a vector normal to the plane. The equation of the plane $2x-y+z=1$ implies that $(2,-1,1)$ is a normal vector to the plane. If you project the vector $(1,1,1)$ onto $(2,-1,1)$, the component of $(1,1,1)$ that was "erased" by this projection is precisely the component lying in the plane. So, $$b- \text{proj}_{(2,-1,1)}(b) = \text{proj}_{2x-y+z=1}(b).$$

Answer 2

Guide for this particular question:

• If a point lies on the plane, then its projection on the plane is the point itself.