(a) Let r(t) and y(t) be differentiable paths in R". Prove the Dot Product Rule: H [x(t) . y(t )] = x'(t) . y(t) + x(t) . y' (t). (b) An n x n matrix A is called skew-symmetric if A"= -A, where A' denotes the transpose of A. Show that if A is a skew-symmetric n x n matrix, then for any two vectors u and v in R" we have Au . v = -u . Av, where . again denotes the dot product. (c) Let A be a skew-symmetric n x n matrix and r(t) be a differentiable path in R" such that I' (t) = Ax(t), for all t. Show that la(t) | | is constant