if u and v are non-collinear vectors, show that u, uxv, and (uxv)xu are mutually orthogonal

By definition the cross product of vectors \vec a and \vec b is orthogonal to the plane containing \vec a and \vec b. So \vec u being orthogonal to \vec u x \vec v is by definition. And (\vec u x \vec v) x \vec u must be orthogonal to both \vec u x \vec v and \vec u .