Find the projection of the point on the plane

Question

I want to find the projection of the point $M(10,-12,12)$ on the plane $2x-3y+4z-17=0$. The normal of the plane is $N(2,-3,4)$.

Do I need to use Gram–Schmidt process? If yes, is this the right formula?

$$\frac{N\cdot M}{|N\cdot N|} \cdot N$$

What will the result be, vector or scalar?

Thanks!

Answer 1

Set the projection point on the plane as $P=(x,y,z)$.

You need three equations:

1. Point $P$ on the plane. $$2x-3y+4z=17$$

2. $\vec{MP}\perp plane$

$$\vec{MP}\perp \vec{PQ_1}$$

$$\vec{MP}\perp \vec{PQ_2}$$

where $Q_1$ and $Q_2$ are two different points on the plane.

Because $\vec{MP}// \vec{N}$, you can use $\vec{N}$ instead of $\vec{MP}$ above.

Answer 2

You can use calculus to minimize the distance (easier: the square of the distance) of M from the generic point of the plane.
Use the equation of the plane to drop a variable, obtaining a function of two independent variables: compute the partial derivatives and find the stationary point. That's all.