# Paper 1, Section II, $5 \mathrm{C}$

Explain why each of the equations

\begin{aligned} &\mathbf{x}=\mathbf{a}+\lambda \mathbf{b} \\ &\mathbf{x} \times \mathbf{c}=\mathbf{d} \end{aligned}

describes a straight line, where $\mathbf{a}, \mathbf{b}, \mathbf{c}$ and $\mathbf{d}$ are constant vectors in $\mathbb{R}^{3}, \mathbf{b}$ and $\mathbf{c}$ are non-zero, $\mathbf{c} \cdot \mathbf{d}=0$ and $\lambda$ is a real parameter. Describe the geometrical relationship of a, $\mathbf{b}, \mathbf{c}$ and $\mathbf{d}$ to the relevant line, assuming that $\mathbf{d} \neq \mathbf{0}$.

Show that the solutions of (2) satisfy an equation of the form (1), defining $\mathbf{a}, \mathbf{b}$ and $\lambda(\mathbf{x})$ in terms of $\mathbf{c}$ and $\mathbf{d}$ such that $\mathbf{a} \cdot \mathbf{b}=0$ and $|\mathbf{b}|=|\mathbf{c}|$. Deduce that the conditions on $\mathbf{c}$ and $\mathbf{d}$ are sufficient for (2) to have solutions.

For each of the lines described by (1) and (2), find the point $\mathbf{x}$ that is closest to a given fixed point $\mathbf{y}$.

Find the line of intersection of the two planes $\mathbf{x} \cdot \mathbf{m}=\mu$ and $\mathbf{x} \cdot \mathbf{n}=\nu$, where $\mathbf{m}$ and $\mathbf{n}$ are constant unit vectors, $\mathbf{m} \times \mathbf{n} \neq \mathbf{0}$ and $\mu$ and $\nu$ are constants. Express your answer in each of the forms (1) and (2), giving both $\mathbf{a}$ and $\mathbf{d}$ as linear combinations of $\mathbf{m}$ and $\mathbf{n}$.