Intersection of line with plane

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have vector equations $$\mathbf { r } = 4 \mathbf { i } - 2 \mathbf { j } + \lambda ( 2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 4 \mathbf { i } - 5 \mathbf { j } + 2 \mathbf { k } + \mu ( \mathbf { i } - \mathbf { j } - \mathbf { k } )$$ respectively.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
2. Find the perpendicular distance from the point \(P\) whose position vector is \(3 \mathbf { i } - 5 \mathbf { j } + 6 \mathbf { k }\) to the plane containing \(l _ { 1 }\) and \(l _ { 2 }\).
3. Find the perpendicular distance from \(P\) to \(l _ { 1 }\).

The line \(l\) has equation \(\mathbf{r} = 5\mathbf{i} - 3\mathbf{j} - \mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + \mathbf{k})\). The plane \(p\) has equation $$(\mathbf{r} - \mathbf{i} - 2\mathbf{j}) \cdot (3\mathbf{i} + \mathbf{j} + \mathbf{k}) = 0.$$ The line \(l\) intersects the plane \(p\) at the point \(A\).

1. Find the position vector of \(A\). [3]
2. Calculate the acute angle between \(l\) and \(p\). [4]
3. Find the equation of the line which lies in \(p\) and intersects \(l\) at right angles. [4]

A line \(l\) has equation \(\frac{x - 6}{-4} = \frac{y + 7}{8} = \frac{z + 10}{7}\) and a plane \(p\) has equation \(3x - 4y - 2z = 8\).

1. Find the point of intersection of \(l\) and \(p\). [3]
2. Find the equation of the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(ax + by + cz = d\). [5]