1995 A-Level H2 Mathematics

Relative to an origin $O$, points $C$ and $D$ have position vectors $7\mathbf{i}+3\mathbf{j}+2\mathbf{k}$ and $10\mathbf{i}+a\mathbf{j}+b\mathbf{k}$ respectively, where $a$ and $b$ are constants.

1. The straight line through $C$ and $D$ has equation $\mathbf{r}=\left( \begin{matrix} 7 \\ 3 \\ 2 \\ \end{matrix} \right)+t\left( \begin{matrix} 1 \\ 3 \\ 0 \\ \end{matrix} \right)$, $t\in \mathbb{R}$. Find the values of $a$ and $b$.
2. Calculate the acute angle between the straight line though $C$ and $D$ and the $y$-axis, giving your correct answer to the nearest ${{0.1}^{\circ }}$.
3. Find the position vector of the point $Q$ on the line $CD$ such that the angle between $\overrightarrow{OQ}$ and $\overrightarrow{OC}$ is equal to the angle between $\overrightarrow{OQ}$ and $\overrightarrow{OD}$ .