# Paper 1, Section II, A

(a) Define the vector product $\mathbf{x} \times \mathbf{y}$ of the vectors $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{R}^{3}$. Use suffix notation to prove that

$\mathbf{x} \times(\mathbf{x} \times \mathbf{y})=\mathbf{x}(\mathbf{x} \cdot \mathbf{y})-\mathbf{y}(\mathbf{x} \cdot \mathbf{x})$

(b) The vectors $\mathbf{x}_{n+1}(n=0,1,2, \ldots)$ are defined by $\mathbf{x}_{n+1}=\lambda \mathbf{a} \times \mathbf{x}_{n}$, where $\mathbf{a}$ and $\mathbf{x}_{0}$ are fixed vectors with $|\mathbf{a}|=1$ and $\mathbf{a} \times \mathbf{x}_{0} \neq \mathbf{0}$, and $\lambda$ is a positive constant.

(i) Write $\mathbf{x}_{2}$ as a linear combination of $\mathbf{a}$ and $\mathbf{x}_{0}$. Further, for $n \geqslant 1$, express $\mathbf{x}_{n+2}$ in terms of $\lambda$ and $\mathbf{x}_{n}$. Show, for $n \geqslant 1$, that $\left|\mathbf{x}_{n}\right|=\lambda^{n}\left|\mathbf{a} \times \mathbf{x}_{0}\right|$.

(ii) Let $X_{n}$ be the point with position vector $\mathbf{x}_{n}(n=0,1,2, \ldots)$. Show that $X_{1}, X_{2}, \ldots$ lie on a pair of straight lines.

(iii) Show that the line segment $X_{n} X_{n+1}(n \geqslant 1)$ is perpendicular to $X_{n+1} X_{n+2}$. Deduce that $X_{n} X_{n+1}$ is parallel to $X_{n+2} X_{n+3}$.

Show that $\mathbf{x}_{n} \rightarrow \mathbf{0}$ as $n \rightarrow \infty$ if $\lambda<1$, and give a sketch to illustrate the case $\lambda=1$.

(iv) The straight line through the points $X_{n+1}$ and $X_{n+2}$ makes an angle $\theta$ with the straight line through the points $X_{n}$ and $X_{n+3}$. Find $\cos \theta$ in terms of $\lambda$.