Let u,v in R3. Define the cross product of u and v to be the vector

u x v=(det [u2 u3 / v2 v3], det [u3 u1 / v3 v1], det [u1 u2/ v1 v2])

Show that u x v is orthogonal to u and v.

I started by taking the dot product of u with u x v and v with u x v, and showing that that needs to be zero. However, I'm having a hard time applying the dot product when I simplify it, and I think I'm just getting stuck on something fairly silly.

I get

u dot(u2v3-u3v2), u dot(u1v3-u3v1), u dot(u1v2-u2v1)

And I'mn ot sure how to proceed.