I want to find the projection of $(2, 3, 8)$ on the normal to the plane $2x_1 + 2x_2 + x_3 = 4$.
I know this question is basically asking me to find the coordinate of the perpendicular from the point to the plane. I've seen several examples but I don't quite understand why signed distance and unit vector is used, as well as how to identify if the vector is in opposite direction.
Could someone please walk me through the method of solving this question? Thank you.
Best Answer
A unit normal vector to the plane is $\frac 1 3 (2,2,1)$. Orthogonal projection of $v$ on a unit vector $w$ is $ \langle v, w \rangle w$. So the projection in this case is $(4,4,2)$.