Vector Cross Product is Orthogonal to Factors

Theorem
Let $\mathbf u$ and $\mathbf v$ be vectors in the Euclidean space $\R^3$.

When $\times$ denotes the vector cross product, we have:


 * $(1): \quad$ $\mathbf u$ and $\mathbf u \times \mathbf v$ are orthogonal.


 * $(2): \quad$ $\mathbf v$ and $\mathbf u \times \mathbf v$ are orthogonal.

Proof
Let $\mathbf u = \left({u_1, u_2, u_3}\right)$, and $\mathbf v = \left({v_1, v_2, v_3}\right)$.

Then $\mathbf u$ and $\mathbf u \times \mathbf v$ are orthogonal, as their dot product is equal to zero:

Similarly, it is shown that $\mathbf v$ and $\mathbf u \times \mathbf v$ are orthogonal: