Review question

# Can we find the equation of the plane containing $l_1$ and $PQ$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5089

## Suggestion

The lines $l_1$ and $l_2$ have equations $\mathbf{r} = \mathbf{a} + s\mathbf{b}$ and $\mathbf{r} = \mathbf{c} + t\mathbf{d}$, where $s$ and $t$ are real numbers and

$\mathbf{a} = 5\mathbf{i}+ 6\mathbf{j}- \mathbf{k}, \quad\mathbf{b} = \mathbf{i} +\mathbf{j}+ \mathbf{k}, \quad\mathbf{c} = \mathbf{i}+ 13\mathbf{j} +7\mathbf{k}, \quad\mathbf{d} = \mathbf{i} + 6\mathbf{j} + 3\mathbf{k}.$

Also, the point $P$ on $l_1$ and the point $Q$ on $l_2$ are such that $PQ$ is perpendicular to both $l_1$ and $l_2.$ In any order:

1. show that $PQ = \sqrt{38},$

2. find the position vectors of $P$ and $Q,$

If $P$ is on $l_1$, what form must its position vector $\mathbf{p}$ take? How about $Q$ with position vector $\mathbf{q}$?

If $PQ$ is perpendicular to both $l_1$ and $l_2$, what can we write down?

1. obtain the Cartesian equation of the plane containing $l_1$ and $PQ$.

Can we find a vector that is normal to the required plane?