Shortest distance from a point to a line Problem set B

The line \(l\) has equation \(\mathbf{r} = \begin{pmatrix}-2\\2\\0\end{pmatrix} + \lambda\begin{pmatrix}2\\1\\0\end{pmatrix}\). The point \(A\) has coordinates \((4, 0, -5)\). The point \(B\) lies on \(l\) and \(AB\) is perpendicular to \(l\).

1. (a) Find the coordinates of \(B\).
2. (b) Hence find the shortest distance from \(A\) to \(l\).

Find the shortest distance from the point \(A(0, 1, 4)\) to the line \(\mathbf{r} = \begin{pmatrix}1\\5\\4\end{pmatrix} + \lambda\begin{pmatrix}0\\-1\\3\end{pmatrix}\).

Find the shortest distance from the point \(P(0, 4, 2)\) to the line \(\mathbf{r} = (4\mathbf{i} - 4\mathbf{j}) + \lambda(2\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})\).

Find the shortest distance from the point \(Q(1, 1, 3)\) to the line \(\mathbf{r} = (\mathbf{j}) + \lambda(\mathbf{j} + 2\mathbf{k})\).

A glider flies along a straight path. Its position vector, in kilometres from a launch field, is \(\mathbf{r} = \begin{pmatrix}3\\5\\3\end{pmatrix} + \lambda\begin{pmatrix}0\\-3\\1\end{pmatrix}\), where \(\lambda\) is the time in hours. A ground marker is at \(A(5, 1, 3)\). Find the shortest distance between the glider's path and the marker, in kilometres.

A helicopter flies along a straight path. Its position vector, in kilometres from a base, is \(\mathbf{r} = \begin{pmatrix}6\\1\\0\end{pmatrix} + \lambda\begin{pmatrix}0\\-1\\2\end{pmatrix}\), where \(\lambda\) is the time in hours after take-off. A beacon is at \(A(1, 4, 2)\). The helicopter is closest to the beacon at the point \(B\).

1. (a) Find the value of \(\lambda\) when the helicopter is at \(B\).
2. (b) Find the shortest distance from \(A\) to the path, in kilometres.

The line \(l\) passes through \(P(2, 1, 0)\) and \(Q(3, -1, 2)\). Find the shortest distance from \(A(4, -2, 5)\) to \(l\).