Sum of three cross products is zero.

Question

Let $u,v,w\in \mathbb R^3$. Prove $u \times( v \times w)+v \times( w \times u)+w \times( u \times v) =0$

I guess things would work out if I just expanded as a ton of products. Is there a better way?

Answer 1

There is a property which should make this question easier:

$$a \times (b \times c) = b(a \cdot c) - c(a \cdot b)$$

You will notice that all the terms cancel out, leading to the answer of $0$ as desired.

Answer 2

$f(x,y,z) = x \times (y \times z) + y \times (z \times x) + z \times (x \times y)$ is linear in each of $x$, $y$, $z$, so it suffices to prove for the cases where each of $x$, $y$ and $z$ is one of the three standard unit vectors $i$, $j$, $k$. Now $f(x,y,z)$ is multiplied by $-1$ if you interchange any two of $x$, $y$ and $z$: in particular if two are equal it must be $0$. So only one case needs to be computed: $$ f(i,j,k) = i \times (j \times k) + j \times (k \times i) + k \times (i \times j)$$ where each of the terms is $0$