Vector Cross Product is Orthogonal to Factors

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

Let $\mathbf a \times \mathbf b$ denote the vector cross product.

Then:


 * $(1): \quad$ $\mathbf a$ and $\mathbf a \times \mathbf b$ are orthogonal.


 * $(2): \quad$ $\mathbf b$ and $\mathbf a \times \mathbf b$ are orthogonal.

Proof
Let $\mathbf a = \begin {bmatrix} a_1 \\ a_2 \\ a_3 \end {bmatrix}$, and $\mathbf b = \begin {bmatrix} b_1 \\ b_2 \\ b_3 \end {bmatrix}$.

Then the dot product of $\mathbf a$ and $\mathbf a \times \mathbf b$ is:

Since the dot product is equal to zero, the vectors $\mathbf a$ and $\mathbf a \times \mathbf b$ are orthogonal by definition.

Similarly, $\mathbf b$ and $\mathbf a \times \mathbf b$ are orthogonal: