Find optimum projection vector under constraints

Question

Given the normal vector $\mathbf{n}$ of a plane in 3 dimensions.

I'm looking for a unit vector $\mathbf{p}$ in that plane for which the dot product of $\mathbf{p}$ with $[1, 1, 1]$ is maximum.

How can I find it?

Answer 1

Take $n = (a,b,c)$ and let $p = (x_p, y_p, z_p)$ be a point in the plane, the plane is given by the vectors $v$ such that: $$ \langle n , v-p \rangle = 0 $$ In other words taking $v = (x,y,z)$: $$ a(x-x_p)+b(y-y_p)+c(z-z_p) = 0$$ Since you are looking for the dot product with $(1,1,1)$ to be maximal, then you want $x+y+z$ to sum the maximum, we can only express one of the variables in terms of the two others, choose $z$: $$ z = -\frac{a(x-x_p) + b(y-y_p) -cz_p}{c}$$ So you want to maximize: $$ f(x,y) = x+y-\frac{a(x-x_p) + b(y-y_p) -cz_p}{c} $$ If you have no other restrictions you can make $x,y \to \infty$ and get $f(x,y)\to\infty$.