Question 1321

Vectors II: Lines and Planes

2021 Paper 1 Question 8 Variant

Question

The line

{l_1}

and

{l_2}

have equations

\begin{align*} & \mathbf{r}_1 = \begin{pmatrix} 2 \\ 1 \\ - 3 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix} \quad \textrm{and} \\ & \mathbf{r}_2 = \begin{pmatrix} - 6 \\ - 4 \\ - 11 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ - 2 \\ 1 \end{pmatrix} \end{align*}

respectively, where

{\lambda}

and

{\mu}

are parameters.

(a)

Find a cartesian equation of the plane containing

{l_1}

and the point

{\left( - 2, - 1, - 1 \right).}

[4]

(b)

Show that

{l_1}

is perpendicular to

{l_2.}

[2]

(ci)

Find the values of

{\lambda}

and

{\mu}

such that

{\mathbf{r}_1 - \mathbf{r}_2}

is perpendicular to both

{l_1}

and

{l_2.}

State the position vectors of the points where the common perpendicular meets

{l_1}

and

{l_2.}

[6]

(cii)

Find the length of this common perpendicular.

[2]