The acute angle between two lines Problem set B

Find the acute angle between the lines\[ l_1:\ \mathbf{r} = \begin{pmatrix}3\\-2\\5\end{pmatrix} + \lambda\begin{pmatrix}2\\3\\-6\end{pmatrix} \]\[ l_2:\ \mathbf{r} = \begin{pmatrix}1\\1\\0\end{pmatrix} + \mu\begin{pmatrix}3\\-4\\12\end{pmatrix} \]

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

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

Find the acute angle between the lines\[ l_1:\ \mathbf{r} = \begin{pmatrix}-4\\1\end{pmatrix} + \lambda\begin{pmatrix}3\\1\end{pmatrix} \]\[ l_2:\ \mathbf{r} = \begin{pmatrix}0\\-6\end{pmatrix} + \mu\begin{pmatrix}1\\5\end{pmatrix} \]

Two straight pipes in a factory are modelled by the lines below. Positions are in metres from a corner of the floor. Find the acute angle between the pipes.\[ l_1:\ \mathbf{r} = \begin{pmatrix}2\\1\\0\end{pmatrix} + \lambda\begin{pmatrix}2\\6\\3\end{pmatrix} \]\[ l_2:\ \mathbf{r} = \begin{pmatrix}0\\4\\5\end{pmatrix} + \mu\begin{pmatrix}-6\\-3\\2\end{pmatrix} \]

Two straight rail tracks on a plan are modelled by the lines below. Distances are in kilometres from the station. Find the acute angle at which the tracks meet.\[ l_1:\ \mathbf{r} = \begin{pmatrix}2\\0\end{pmatrix} + \lambda\begin{pmatrix}8\\15\end{pmatrix} \]\[ l_2:\ \mathbf{r} = \begin{pmatrix}10\\-4\end{pmatrix} + \mu\begin{pmatrix}4\\-3\end{pmatrix} \]

The line \(l_1\) passes through \(P(0, 3, 2)\) and \(Q(2, -1, 8)\). The line \(l_2\) has equation \(\mathbf{r} = \begin{pmatrix}1\\0\\5\end{pmatrix} + \mu\begin{pmatrix}-4\\-1\\2\end{pmatrix}\). Find the acute angle between \(l_1\) and \(l_2\).

The lines below are perpendicular. Find the value of \(k\).\[ l_1:\ \mathbf{r} = \begin{pmatrix}2\\1\\4\end{pmatrix} + \lambda\begin{pmatrix}2\\k\\-1\end{pmatrix} \]\[ l_2:\ \mathbf{r} = \begin{pmatrix}4\\0\\2\end{pmatrix} + \mu\begin{pmatrix}1\\2\\5\end{pmatrix} \]