Parametric form, intersecting lines and skew lines Problem set A

Write the parametric equations of the line \(\mathbf{r}=\begin{pmatrix}-2\\5\\1\end{pmatrix}+\lambda\begin{pmatrix}3\\0\\-4\end{pmatrix}\).

A line has parametric equations \(x=4-2t\), \(y=1+5t\), \(z=3\).

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

Find the coordinates of the point of intersection of the lines \(\mathbf{r}_1=\begin{pmatrix}-3\\5\\-1\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-1\end{pmatrix}\) and \(\mathbf{r}_2=\begin{pmatrix}2\\5\\0\end{pmatrix}+\mu\begin{pmatrix}1\\-2\\3\end{pmatrix}\).

Show that the lines \(\mathbf{r}_1=\begin{pmatrix}0\\3\\-1\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\2\end{pmatrix}\) and \(\mathbf{r}_2=\begin{pmatrix}5\\-2\\4\end{pmatrix}+\mu\begin{pmatrix}2\\-1\\1\end{pmatrix}\) do not intersect.

Two drones fly along straight paths with equations \(\mathbf{r}_1=\begin{pmatrix}10\\42\\24\end{pmatrix}+\lambda\begin{pmatrix}5\\2\\-1\end{pmatrix}\) and \(\mathbf{r}_2=\begin{pmatrix}45\\30\\15\end{pmatrix}+\mu\begin{pmatrix}-3\\4\\1\end{pmatrix}\), with coordinates in metres.

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

The flight paths of two aircraft are modelled by the lines \(\mathbf{r}_1=\begin{pmatrix}0\\0\\9\end{pmatrix}+\lambda\begin{pmatrix}4\\3\\-1\end{pmatrix}\) and \(\mathbf{r}_2=\begin{pmatrix}10\\-5\\7\end{pmatrix}+\mu\begin{pmatrix}1\\2\\0\end{pmatrix}\), with coordinates in kilometres.

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

On a map, two straight roads have equations \(\mathbf{r}_1=\begin{pmatrix}1\\2\end{pmatrix}+\lambda\begin{pmatrix}3\\1\end{pmatrix}\) and \(\mathbf{r}_2=\begin{pmatrix}10\\-2\end{pmatrix}+\mu\begin{pmatrix}-1\\2\end{pmatrix}\), with distances in kilometres. Find the coordinates of the point where the roads meet.

The lines \(\mathbf{r}_1=\begin{pmatrix}1\\0\\2\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-1\end{pmatrix}\) and \(\mathbf{r}_2=\begin{pmatrix}3\\-1\\0\end{pmatrix}+\mu\begin{pmatrix}1\\-1\\2\end{pmatrix}\) 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 \(\mathbf{r}_1=\begin{pmatrix}1\\-1\\k\end{pmatrix}+\lambda\begin{pmatrix}2\\1\\-1\end{pmatrix}\quad\text{and}\quad \mathbf{r}_2=\begin{pmatrix}5\\4\\5\end{pmatrix}+\mu\begin{pmatrix}1\\-1\\2\end{pmatrix}\) intersect. Find the value of \(k\) and the coordinates of the point of intersection.