In this problem, we discuss the cross product, which is a product of two vectors in R^3 that gives another vector. This product exists only in R^3, not in higher dimensions. Let u = (u1, u2, u3) and v = (v1, v2, v3). Then we define u x v = (u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1). (Often, the heuristic u x v = |e1 e2 e3| |u1 u2 u3| |v1 v2 v3| is used). 1. Show that for any vector w, the equation w · (u x v) = det(w, u, v) holds, where the determinant is of the matrix are the three given columns. 2. Show that u x v is orthogonal to both u and v. (This fact is one of the main things that makes the cross product so useful: any time you have two vectors and you want to find a third vector that is orthogonal to both, the cross product is by far the easiest tool to use.)