Question 7 13 marks Paper 2 Section A Hard Calculator Free
Maximum mark: 13
Consider the points \( P(-3, -1, 2) \) and \( Q(5, 5, 6) \).
(a) Find a vector equation for the line, \( L_{1} \), which passes through the points \( P \) and \( Q \).
The line \( L_{2} \) has equation
\[ \mathbf{r} = \begin{pmatrix} -4 \\ 0 \\ 4 \end{pmatrix} + s\begin{pmatrix} 5 \\ 2 \\ 0 \end{pmatrix}. \]
[2]
(b) Show that \( L_{1} \) and \( L_{2} \) intersect at the point \( R(1, 2, 4) \).[2]
(c) Find the acute angle between \( L_{1} \) and \( L_{2} \).
Let \( S \) be a point on \( L_{2} \) such that \( \left|RP\right| = \left|RS\right| \).[3]
(d) Show that one of the possible positions for \( S \) is \( S_{1}(-4, 0, 4) \) and find the coordinates of the other possible position, \( S_{2} \).[3]
(e) Find a vector equation of the line which passes through \( R \) and bisects \( \widehat{PRS_{1}} \).[3]
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