To solve this problem, we exploit the fact that we have two different ways of computing the dot product. On the one hand, using the component method, we know that v·w = 2 - 10 + 12 = 4 . On the other hand, we know from the geometric method that v·w = | v|| w| cosθ . From the components we can compute | v|² = 4 + 25 + 9 = 38 , and | w|² = 1 + 4 + 16 = 21 . Putting all of these equations together we find that

We know from the geometric formula that the dot product between two perpendicular vectors is zero. Hence we are looking for a vector (a, b, c) such that if we dot it into either u or v we get zero. This gives us two equations:

Any choice of a , b , and c which satisfies these equations works. One possible answer is the vector (2, 1, - 3) , but any scalar multiple of this vector will also be perpendicular to u and v .