Problem

Relative to a fixed origin $O$, the lines $l_1$ and $l_2$ are given by the equations

$$l_1:\ \mathbf{r}= \begin{pmatrix} 4\\-3\\2 \end{pmatrix} +\lambda \begin{pmatrix} 3\\-2\\-1 \end{pmatrix} \qquad \text{where }\lambda\text{ is a scalar parameter}$$ $$l_2:\ \mathbf{r}= \begin{pmatrix} 2\\0\\-9 \end{pmatrix} +\mu \begin{pmatrix} 2\\-1\\-3 \end{pmatrix} \qquad \text{where }\mu\text{ is a scalar parameter}$$

Given that $l_1$ and $l_2$ meet at the point $X$,

(a) find the position vector of $X$.

The point $P(10,-7,0)$ lies on $l_1$

The point $Q$ lies on $l_2$

Given that $\overrightarrow{PQ}$ is perpendicular to $l_2$

(b) calculate the coordinates of $Q$.