# Paper 1, Section II, 5B

(i) For vectors $\mathbf{a}, \mathbf{b}, \mathbf{c} \in \mathbb{R}^{3}$, show that

$\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \cdot \mathbf{c}) \mathbf{b}-(\mathbf{a} \cdot \mathbf{b}) \mathbf{c} .$

Show that the plane $(\mathbf{r}-\mathbf{a}) \cdot \mathbf{n}=0$ and the line $(\mathbf{r}-\mathbf{b}) \times \mathbf{m}=\mathbf{0}$, where $\mathbf{m} \cdot \mathbf{n} \neq 0$, intersect at the point

$\mathbf{r}=\frac{(\mathbf{a} \cdot \mathbf{n}) \mathbf{m}+\mathbf{n} \times(\mathbf{b} \times \mathbf{m})}{\mathbf{m} \cdot \mathbf{n}}$

and only at that point. What happens if $\mathbf{m} \cdot \mathbf{n}=0$ ?

(ii) Explain why the distance between the planes $\left(\mathbf{r}-\mathbf{a}_{1}\right) \cdot \hat{\mathbf{n}}=0$ and $\left(\mathbf{r}-\mathbf{a}_{2}\right) \cdot \hat{\mathbf{n}}=0$ is $\left|\left(\mathbf{a}_{1}-\mathbf{a}_{2}\right) \cdot \hat{\mathbf{n}}\right|$, where $\hat{\mathbf{n}}$ is a unit vector.

(iii) Find the shortest distance between the lines $(3+s, 3 s, 4-s)$ and $(-2,3+t, 3-t)$ where $s, t \in \mathbb{R}$. [You may wish to consider two appropriately chosen planes and use the result of part (ii).]