[Math] The orthogonal projection onto a plane – explanation

Question

Could somebody explain, why orthogonal projection onto a plane with equation $x_1+x_2+x_3=0$ is given by $$y=(x_1,x_2,x_3)-\bigg( \frac{x_1+x_2+x_3}{3}\bigg)(1,1,1)$$

I don't understand, why we sum three coordinates and divide by $3$? I thought, we need to use the dot product of the point and the normal instead of taking weighted average, so I'm a bit confused?

I suppose this has to do with the geometry of how the plane sits in the 3D space and arbitrary point. We basically need to find the distance of arbitrary point to the the plane, but I still don't see how this translates into the given form.

Thanks!

Answer 1

Notice that the unit normal to your plane $x_1+x_2+x_3=0$ is $\frac1{\sqrt3}(1,1,1)$. Use the dot product formula with this unit normal and you'll get the formula in your question.