Exercise 21

One of the properties of the cross product is that $(\vu+\vv) \times \vw = (\vu \times \vw) + (\vv \times \vw)$. That is, the cross product distributes over vector addition on the right. Here we investigate whether the cross product distributes over vector addition on the left.

1. Let $\vu = \langle 1,2,-1 \rangle$, $\vv = \langle 4,-3,6 \rangle$, and $\vw = \langle 4,7,2 \rangle$. Calculate $$\begin{gathered} \vu \times (\vv + \vw) \ \ \text{ and } \ \ (\vu \times \vv) + (\vu \times \vw). \end{gathered}$$ What do you notice?

2. Use the properties of the cross product to show that in general $$\begin{gathered} \vx \times (\vy + \vz) = (\vx \times \vy) + (\vx \times \vz) \end{gathered}$$ for any vectors $\vx$, $\vy$, and $\vz$ in $\R^3$.