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.
