Review question

# Where are $P$ and $Q$ if $PQ$ is perpendicular to $l_1$ and $l_2$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8828

## Question

The equations of the lines $l_1$ and $l_2$ are $\mathbf{r}=\mathbf{a}+\lambda \mathbf{b}$ and $\mathbf{r}=\mathbf{c}+\mu \mathbf{d}$ respectively, where $\mathbf{a}=-6\mathbf{i}+3\mathbf{j}+15\mathbf{k}, \hspace{0.5cm} \mathbf{b}=\mathbf{i}-2\mathbf{j}+3\mathbf{k}, \hspace{0.5cm} \mathbf{c}=6\mathbf{i}+15\mathbf{j}+39\mathbf{k}, \hspace{0.5cm} \mathbf{d}=2\mathbf{i}-3\mathbf{j}+4\mathbf{k},$

and where $\lambda$ and $\mu$ are scalar parameters. The points $P$ and $Q$ are on $l_1$ and $l_2$ respectively, and $PQ$ is perpendicular to both $l_1$ and $l_2$. In any order:

1. find a vector which is parallel to $\overrightarrow{PQ}$, giving your answer in the form $x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$;
2. show that $PQ=10\sqrt{6}$;
3. find the position vectors of $P$ and $Q$.