Parametric form, intersecting lines and skew lines Problem set B

Write the parametric equations of the line \(\mathbf{r}=\begin{pmatrix}6\\-1\\0\end{pmatrix}+\lambda\begin{pmatrix}0\\-2\\5\end{pmatrix}\).

A line has parametric equations \(x=-1+4s\), \(y=3s\), \(z=7-2s\).

1. (a) Write down a vector equation of the line.
2. (b) Find the coordinates of the point on the line where \(s=3\).

Find the coordinates of the point of intersection of the lines \(\mathbf{r}_1=\begin{pmatrix}-7\\11\\0\end{pmatrix}+\lambda\begin{pmatrix}1\\-3\\2\end{pmatrix}\) and \(\mathbf{r}_2=\begin{pmatrix}-8\\0\\8\end{pmatrix}+\mu\begin{pmatrix}2\\1\\-1\end{pmatrix}\).

Show that the lines \(\mathbf{r}_1=\begin{pmatrix}2\\-4\\1\end{pmatrix}+\lambda\begin{pmatrix}3\\1\\-2\end{pmatrix}\) and \(\mathbf{r}_2=\begin{pmatrix}-1\\0\\3\end{pmatrix}+\mu\begin{pmatrix}1\\2\\1\end{pmatrix}\) do not intersect.

Two pipes are laid underground along the lines \(\mathbf{r}_1=\begin{pmatrix}4\\-8\\-1\end{pmatrix}+\lambda\begin{pmatrix}1\\3\\-2\end{pmatrix}\) and \(\mathbf{r}_2=\begin{pmatrix}0\\1\\-8\end{pmatrix}+\mu\begin{pmatrix}2\\-1\\1\end{pmatrix}\), with coordinates in metres.

1. (a) Show that the two pipes cross.
2. (b) Find the coordinates of the crossing point.

Two cable-car routes are modelled by the lines \(\mathbf{r}_1=\begin{pmatrix}1\\1\\6\end{pmatrix}+\lambda\begin{pmatrix}2\\-1\\1\end{pmatrix}\) and \(\mathbf{r}_2=\begin{pmatrix}7\\0\\2\end{pmatrix}+\mu\begin{pmatrix}3\\2\\-1\end{pmatrix}\), with coordinates in kilometres.

1. (a) Show that the two routes do not cross.
2. (b) Explain why this does not mean they are parallel.

On a map, two straight railway lines have equations \(\mathbf{r}_1=\begin{pmatrix}-8\\14\end{pmatrix}+\lambda\begin{pmatrix}2\\-3\end{pmatrix}\) and \(\mathbf{r}_2=\begin{pmatrix}-4\\1\end{pmatrix}+\mu\begin{pmatrix}1\\2\end{pmatrix}\), with distances in kilometres. Find the coordinates of the point where the railway lines meet.

The lines \(\mathbf{r}_1=\begin{pmatrix}0\\2\\-1\end{pmatrix}+\lambda\begin{pmatrix}1\\3\\2\end{pmatrix}\) and \(\mathbf{r}_2=\begin{pmatrix}4\\1\\3\end{pmatrix}+\mu\begin{pmatrix}2\\-1\\1\end{pmatrix}\) are in three dimensions.

1. (a) Show that the lines are not parallel.
2. (b) Show that the lines do not intersect.
3. (c) Write down the word for two lines that are neither parallel nor intersecting.

The lines \(\mathbf{r}_1=\begin{pmatrix}-5\\1\\k\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\3\end{pmatrix}\quad\text{and}\quad \mathbf{r}_2=\begin{pmatrix}-11\\9\\0\end{pmatrix}+\mu\begin{pmatrix}2\\-1\\1\end{pmatrix}\) intersect. Find the value of \(k\) and the coordinates of the point of intersection.