If u+v and u-v are orthogonal and the length of u is 1. Find the length of v.
Use the cross product to show that if u+v and u-v are parallel, then u and v are also parallel.
Let * denote the dot product. Since u+v and u-v are orthogonal we have (u+v)(u-v)=0 and since the dot product is distributive and commutative we may deduce uu-v*v=0 and thus |u|^2 - |v|^2 = 0, that is |v|=1 since |u|=1.
If they are parallel and x denotes the cross product, that means (u+v)x(u-v)=0 which is uxu - uxv +vxu -vxv = 0. Since axb=-bxa and axa=0 we have 2*vxu=0 that is vxu=0 that is v and u are parallel.