let $\mathbf{x}$ and $\mathbf{y}$ be two perpendicular unit vectors. (a) find a formula for the magnitude of an arbitrary linear combination \[a\mathbf{x} b\mathbf{y}\]of $\mathbf{x}$ and $\mathbf{y}$ in terms of $a$ and $b$. (b) find the precise condition on $a, b, c$ and $d$ under which the linear combinations \[a\mathbf{x} b\mathbf{y} \text{ and } c\mathbf{x} d \mathbf{y}\]are perpendicular. a purely geometric proof will not receive full credit; try to use vectors!