Equation of plane containing line and point/parallel to vector

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = - 5 \mathbf { j } + \lambda ( 5 \mathbf { i } + 2 \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } + \mu ( \mathbf { j } + \mathbf { k } )\) respectively. The plane \(\Pi\) contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).

1. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the distance between \(l _ { 2 }\) and \(\Pi\).
   The point \(P\) on \(l _ { 1 }\) and the point \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\).
3. Show that \(P\) has position vector \(\frac { 55 } { 27 } \mathbf { i } - 5 \mathbf { j } + \frac { 22 } { 27 } \mathbf { k }\) and state a vector equation for \(P Q\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = - \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + s ( 2 \mathbf { i } - 3 \mathbf { j } )\) and \(\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { k } + t ( 3 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } )\) respectively. The plane \(\Pi _ { 1 }\) contains \(l _ { 1 }\) and the point \(P\) with position vector \(- 2 \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k }\).

1. Find an equation of \(\Pi _ { 1 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\).
   The plane \(\Pi _ { 2 }\) contains \(l _ { 2 }\) and is parallel to \(l _ { 1 }\).
2. Find an equation of \(\Pi _ { 2 }\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\).
3. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
4. The point \(Q\) is such that \(\overrightarrow { \mathrm { OQ } } = - 5 \overrightarrow { \mathrm { OP } }\). Find the position vector of the foot of the perpendicular from the point \(Q\) to \(\Pi _ { 2 }\).

Two lines have equations $$\frac{x-k}{2} = \frac{y+1}{-5} = \frac{z-1}{-3} \quad \text{and} \quad \frac{x-k}{1} = \frac{y+4}{-4} = \frac{z}{-2},$$ where \(k\) is a constant.

1. Show that, for all values of \(k\), the lines intersect, and find their point of intersection in terms of \(k\). [6]
2. For the case \(k = 1\), find the equation of the plane in which the lines lie, giving your answer in the form \(ax + by + cz = d\). [4]