Suppose that a,b, $\mathbf{c}, \mathbf{d}$ are the vertices of a regular tetrahedron $T$ in $\mathbb{R}^{3}$ and that $\mathbf{a}=(1,1,1), \mathbf{b}=(-1,-1,1), \mathbf{c}=(-1,1,-1), \mathbf{d}=(1, x, y)$.

(a) Find $x$ and $y$.

(b) Find a matrix $M$ that is a rotation leaving $T$ invariant such that $M \mathbf{a}=\mathbf{b}$ and $M \mathbf{b}=\mathbf{a} .$