Arbitrary Dot and Cross Products

Question

I am having a bit of trouble with answering these few dot and cross product questions.

Suppose that $u · (v × w) =3$. Find,

$w · (u × v)$

$v · (u × w)$

$(u × w) · v$

Could some explain their reasoning behind their answers?

Answer 1

Graphically, all the expressions mentioned are the triple products of $\vec u$,$\vec v$, $\vec w$, which represent the SIGNED volume of the identical parallelepipe constructed by the $3$ vectors. So they have the same value, that is, $3$.

If that is not obvious enough, take the first expression as an example, $v×w$ is the area of the base. While the height is $|\vec u|\cos\theta$, where $\theta$ is the angle between $\vec u$ and the normal vector defined by $\vec v$ and $\vec w$. So the volume of the parallelepipe is $|\vec u|\cos\theta\cdot (v×w)=\vec u\cdot (v×w)$ .

Answer 2

If $u = (u_1, u_2, u_3), \; v = (v_1, v_2, v_3), \; \text{ and } \; w = (w_1, w_2, w_3)$

then $u \cdot (v \times w) = \left| \begin{matrix} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \\ \end{matrix} \right| = 3$

So, for example $w \cdot (u \times v) = \left| \begin{matrix} w_1 & w_2 & w_3 \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ \end{matrix} \right| = -\left| \begin{matrix} w_1 & w_2 & w_3 \\ v_1 & v_2 & v_3 \\ u_1 & u_2 & u_3 \\ \end{matrix} \right| = \left| \begin{matrix} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \\ \end{matrix} \right| =3$