Let $A$ be an orthogonal matrix, i.e. $A^T=A^{-1}$. Prove that $A(x\times{y})=\det(A)(Ax\times{Ay})$. I know that orthogonal matrices preserve distance, angles and orthogonality of vectors and I have tried using this idea in conjunction with the formula $|x\times{y}|=|x||y|sin\theta(x,y)$.
(Side-question: How do we know orthogonal matrices preserve distances, angles and orthogonalit of vectors?)
