Exercise 16

Suppose that $\vec{v}_{1},\vec{v}_{2}\in\mathbb{R}^{n}$ are such that for all complements $M,N\subseteq\mathbb{R}^{n}$, the projections of $\vec{v}_{1}$ and $\vec{v}_{2}$ into $M$ along $N$ are equal. Must $\vec{v}_{1}$ equal $\vec{v}_{2}$? (If so, what if we relax the condition to: all orthogonal projections of the two are equal?)