Suppose that $W$ consists of all vectors $[x, y, z]^T$ such that $x+3y-2z=0.$
What is the distance from $v = [1, 2, 0]^T$ to $W$?
I know that the distance from the vector to the plane is the magnitude of the vector $v$ minus the projection of $v$ on $w$, but I am having trouble coming up with the orthonormal basis needed to use that equation.
A part of me also thinks I might be over complicating this problem... any help is appreciated.
Also, the answer given to me is $sqrt(14)/2$.
Guide:
Rather than finding the orthonormal basis, a faster way is to note that $(1,3,-2)$ is the normal direction, normalize it, project $v$ onto the normal direction and compute its length.
As you mentioned, you have trouble coming up with the orthonormal basis, note that $(-3,1,0)$ is a vector on the plane, $(2,0,1)$ is also another vector on the plane,we can apply Gram-Schmidt algorithm on these two vectors to find an orthonormal basis.