Given a vector v in R^3, I was wondering why summing v's projections onto any two orthogonal vectors in a fixed plane yields the same result, from a geometric point of view.
Using the plane z = 0 as an example, I would hypothesize that
proj v onto [0, 1, 0] + proj v onto [1, 0, 0]
would yield a different vector (most likely in the opposite quadrant of the plane) than
proj v onto [0, -1, 0] + proj v onto [-1, 0, 0]
and that you would get a different vector still for
proj v onto [1, 2, 0] + proj v onto [2, -1, 0]
but when you crunch the numbers, these three calculations are the same. Why is this the case?