Perpendicular distance point to plane

7 The straight line \(l\) has equation \(\mathbf { r } = 4 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } + \lambda ( 2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k } )\). The plane \(p\) passes through the point \(( 4 , - 1,2 )\) and is perpendicular to \(l\).

1. Find the equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the perpendicular distance from the origin to \(p\).
3. A second plane \(q\) is parallel to \(p\) and the perpendicular distance between \(p\) and \(q\) is 14 units. Find the possible equations of \(q\).

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have vector equations $$\mathbf { r } = a \mathbf { i } + 9 \mathbf { j } + 13 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 3 \mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k } + \mu ( - \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } )$$ respectively. It is given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.

1. Find the value of the constant \(a\).
   The point \(P\) has position vector \(3 \mathbf { i } + \mathbf { j } + 6 \mathbf { k }\).
2. Find the perpendicular distance from \(P\) to the plane containing \(l _ { 1 }\) and \(l _ { 2 }\).
3. Find the perpendicular distance from \(P\) to \(l _ { 2 }\).

6 The equation of the plane \(\Pi\) is \(\mathbf { r } = \left( \begin{array} { r } - 1 \\ 2 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { l } 4 \\ 4 \\ 3 \end{array} \right) + \mu \left( \begin{array} { r } - 2 \\ 3 \\ 1 \end{array} \right)\).

1. Find the acute angle between \(\Pi\) and the plane with equation \(\mathbf { r } . \left( \begin{array} { l } 2 \\ 0 \\ 3 \end{array} \right) = 4\). The point \(A\) has coordinates ( \(9 , - 7,20\) ).
   The point \(F\) is the point of intersection between \(\Pi\) and the perpendicular from \(A\) to \(\Pi\).
2. Determine the coordinates of \(F\).

1. The plane \(\Pi\) has equation

$$\mathbf { r } = \left( \begin{array} { l } 3 \\ 3 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 2 \\ 1 \end{array} \right) + \mu \left( \begin{array} { l } 2 \\ 0 \\ 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that vector \(2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\) is perpendicular to \(\Pi\).
2. Hence find a Cartesian equation of \(\Pi\). The line \(l\) has equation $$\mathbf { r } = \left( \begin{array} { r } 4 \\ - 5 \\ 2 \end{array} \right) + t \left( \begin{array} { r } 1 \\ 6 \\ - 3 \end{array} \right)$$ where \(t\) is a scalar parameter.
   The point \(A\) lies on \(l\).
   Given that the shortest distance between \(A\) and \(\Pi\) is \(2 \sqrt { 29 }\)
3. determine the possible coordinates of \(A\).