Question 1309

Vectors II: Lines and Planes

2009 Paper 1 Question 10 Variant

Question

The planes

{p_1}

and

{p_2}

have equations

{\mathbf{r} \cdot \begin{pmatrix} 1 \\ - 1 \\ - 1 \end{pmatrix} = 3}

and

{\mathbf{r} \cdot \begin{pmatrix} - 1 \\ 1 \\ - 2 \end{pmatrix} = - 2}

respectively, and meet in a line

{l.}

(i)

Find the acute angle between

{p_1}

and

{p_2.}

[3]

(ii)

Find a vector equation of

{l.}

[4]

(iii)

The plane

{p_3}

has equation

{x - y - z - 3} + {k(- x + y - 2 z + 2)} = 0.

Explain why

{l}

lies in

{p_3}

for any constant

{k.}

Hence, or otherwise, find a cartesian equation of the plane in which both

{l}

and the point

{\left( 1, - 2, - 3 \right)}

lie.

[5]