Find foot of perpendicular from point to line

10 The points \(A\) and \(B\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O A } = \left( \begin{array} { r } - 1 \\ 3 \\ 5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 3 \\ - 1 \\ - 4 \end{array} \right) .$$ The line \(l\) passes through \(A\) and is parallel to \(O B\). The point \(N\) is the foot of the perpendicular from \(B\) to \(l\).

1. State a vector equation for the line \(l\).
2. Find the position vector of \(N\) and show that \(B N = 3\).
3. Find the equation of the plane containing \(A , B\) and \(N\), giving your answer in the form \(a x + b y + c z = d\).

11 The points \(A\) and \(B\) have position vectors \(\mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }\) and \(2 \mathbf { i } - \mathbf { j } + \mathbf { k }\) respectively. The line \(l\) has equation \(\mathbf { r } = \mathbf { i } - \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k } )\).

1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
2. Find the position vector of the foot of the perpendicular from \(A\) to \(l\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 Points \(A , B\) and \(C\) have coordinates ( \(4,2,0\) ), ( \(1,5,3\) ) and ( \(1,4 , - 2\) ) respectively. The line \(l\) passes through \(A\) and \(B\).

1. Find a cartesian equation for \(l\). \(M\) is the point on \(l\) that is closest to \(C\).
2. Find the coordinates of \(M\).
3. Find the exact area of the triangle \(A B C\).

10 The points \(A ( 5 , - 4,6 )\) and \(B ( 6 , - 6,8 )\) lie on the line \(L\). The point \(C\) is \(( 15 , - 5,9 )\). 10

1. \(D\) is the point on \(L\) that is closest to \(C\).
   Find the coordinates of \(D\).
   10
2. Hence find, in exact form, the shortest distance from \(C\) to \(L\).