Question

U= [2,-5,-1] V=[3,2,-3] Find the orthogonal projection of u onto v. Then write u as the sum of two orthogonal vectors, one in span{U} and one orthogonal to U

Answer 1

There is a mistake. The question should read “Find the orthogonal projection of v onto u. Then write v as the sum of two orthogonal vectors, one in span{u} and one orthogonal to u.

We have proj_v (u) = [(v.u)/(u.u)]u = [(6-10+3)/(4+25+1)]u = -(1/30)u = -(1/30) [2,-5,-1] = [-1/15,1/6,1/30]. This vector, being a scalar multiple of u is in span{u}.

Let x = [-1/15,1/6,1/30]. Then v-x = [3,2,-3]- [-1/15,1/6,1/30]= [ 46/15,11/6,-91/30] = y (say).

Then v = x+y, where x = [-1/15,1/6,1/30] is in span{u} and y = [ 46/15,11/6,-91/30] is orthogonal to u.