The acute angle between two lines Problem set A

Find the acute angle between the lines\[ l_1:\ \mathbf{r} = \begin{pmatrix}1\\4\\-2\end{pmatrix} + \lambda\begin{pmatrix}2\\-1\\2\end{pmatrix} \]\[ l_2:\ \mathbf{r} = \begin{pmatrix}0\\3\\1\end{pmatrix} + \mu\begin{pmatrix}1\\2\\-2\end{pmatrix} \]

Find the acute angle between the lines\[ l_1:\ \mathbf{r} = \begin{pmatrix}0\\-1\\4\end{pmatrix} + \lambda\begin{pmatrix}2\\3\\-1\end{pmatrix} \]\[ l_2:\ \mathbf{r} = \begin{pmatrix}3\\2\\-5\end{pmatrix} + \mu\begin{pmatrix}4\\0\\-3\end{pmatrix} \]

Find the acute angle between the lines\[ l_1:\ \mathbf{r} = \begin{pmatrix}1\\-1\end{pmatrix} + \lambda\begin{pmatrix}3\\2\end{pmatrix} \]\[ l_2:\ \mathbf{r} = \begin{pmatrix}-2\\6\end{pmatrix} + \mu\begin{pmatrix}-4\\1\end{pmatrix} \]

Find the acute angle between the lines\[ l_1:\ \mathbf{r} = \begin{pmatrix}0\\7\end{pmatrix} + \lambda\begin{pmatrix}2\\5\end{pmatrix} \]\[ l_2:\ \mathbf{r} = \begin{pmatrix}-3\\2\end{pmatrix} + \mu\begin{pmatrix}4\\1\end{pmatrix} \]

Two straight cables on a building site are modelled by the lines below. Positions are in metres from a fixed point. Find the acute angle between the cables.\[ l_1:\ \mathbf{r} = \begin{pmatrix}0\\0\\10\end{pmatrix} + \lambda\begin{pmatrix}6\\2\\-3\end{pmatrix} \]\[ l_2:\ \mathbf{r} = \begin{pmatrix}4\\5\\0\end{pmatrix} + \mu\begin{pmatrix}3\\-2\\6\end{pmatrix} \]

Two straight roads on a map are modelled by the lines below. Distances are in kilometres from the town hall. Find the acute angle at which the roads meet.\[ l_1:\ \mathbf{r} = \begin{pmatrix}1\\3\end{pmatrix} + \lambda\begin{pmatrix}3\\-4\end{pmatrix} \]\[ l_2:\ \mathbf{r} = \begin{pmatrix}8\\1\end{pmatrix} + \mu\begin{pmatrix}5\\12\end{pmatrix} \]

The line \(l_1\) passes through \(P(1, 2, -1)\) and \(Q(4, 0, 5)\). The line \(l_2\) has equation \(\mathbf{r} = \begin{pmatrix}2\\2\\0\end{pmatrix} + \mu\begin{pmatrix}1\\-2\\2\end{pmatrix}\). Find the acute angle between \(l_1\) and \(l_2\).

The lines below are perpendicular. Find the value of \(k\).\[ l_1:\ \mathbf{r} = \begin{pmatrix}2\\1\\4\end{pmatrix} + \lambda\begin{pmatrix}k\\2\\1\end{pmatrix} \]\[ l_2:\ \mathbf{r} = \begin{pmatrix}0\\2\\-1\end{pmatrix} + \mu\begin{pmatrix}1\\3\\-4\end{pmatrix} \]