## Problem on the commutativity of the cross product

What is the relationship of $\bfa \times \bfb$ and $\bfb \times \bfa$? Prove it using components.
• ## Solution

Whatever the relationship is, we will need a proof in terms of components. Hence, we write $\bfa \times \bfb$ and $\bfb \times \bfa$ in terms of the components of $\bfa$ and $\bfb$.
Let $\bfa = \langle a_1, a_2, a_3 \rangle.$ Let $\bfb = \langle b_1, b_2, b_3 \rangle.$
Now, let's compute $\bfa \times \bfb$.
Recall that
Applying the same formula for the cross product $\bfb \times \bfa$:
Each component of $\bfa \times \bfb$ is the negative of the corresponding component of $\bfb \times \bfa$.
Hence, we conclude that $\bfa \times \bfb = - ( \bfb \times \bfa)$.