Find foot of perpendicular from point to line

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 \(l\).
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Relative to the origin \(O\), the point \(A\) has position vector given by \(\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 4\mathbf{k}\). The line \(l\) has equation \(\mathbf{r} = 9\mathbf{i} - \mathbf{j} + 8\mathbf{k} + \mu(3\mathbf{i} - \mathbf{j} + 2\mathbf{k})\).

1. Find the position vector of the foot of the perpendicular from \(A\) to \(l\). Hence find the position vector of the reflection of \(A\) in \(l\). [5]
2. Find the equation of the plane through the origin which contains \(l\). Give your answer in the form \(ax + by + cz = d\). [3]
3. Find the exact value of the perpendicular distance of \(A\) from this plane. [3]

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

1. \(D\) is the point on \(L\) that is closest to \(C\). Find the coordinates of \(D\). [6 marks]
2. Hence find, in exact form, the shortest distance from \(C\) to \(L\). [2 marks]

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 \(l\). [3]

\(M\) is the point on \(l\) that is closest to \(C\).

1. Find the coordinates of \(M\). [4]
2. Find the exact area of the triangle \(ABC\). [4]