Question Type 3: Finding the value of parameters for which the two lines would be under a specific classification Bootcamps

Determine the value of $m$ for which the lines $L_1: (2,3,0)+t(1,1,m) \quad\text{and}\quad L_2: (0,4,1)+s(3,1,2)$

are parallel.

Find the value(s) of $m$ such that the lines $\ell_1: \; \mathbf{r} = \begin{pmatrix}1\\2\\3\end{pmatrix} + t\begin{pmatrix}2\\-1\\m\end{pmatrix}, \quad \ell_2: \; \mathbf{r} = \begin{pmatrix}0\\1\\1\end{pmatrix} + s\begin{pmatrix}1\\2\\-3\end{pmatrix}$ are perpendicular.

In 2D, find $k$ such that the lines $\ell_1:\; \mathbf{r} = \begin{pmatrix}2\\-1\end{pmatrix} + t\begin{pmatrix}1\\k\end{pmatrix}, \quad \ell_2:\; \mathbf{r} = \begin{pmatrix}0\\3\end{pmatrix} + s\begin{pmatrix}2\\-1\end{pmatrix}$ are perpendicular.

Determine the value of $m$ such that the 2D lines $\ell_1:\; \mathbf{r} = \begin{pmatrix}1\\ m\end{pmatrix} + t\begin{pmatrix}2\\-3\end{pmatrix}, \quad \ell_2:\; \mathbf{r} = \begin{pmatrix}5\\-7\end{pmatrix} + s\begin{pmatrix}-4\\6\end{pmatrix}$ represent the same line (are coincident).

Find all real $m$ such that the lines $\ell_1: \; \mathbf{r} = \begin{pmatrix}2\\m\\1\end{pmatrix} + t\begin{pmatrix}3\\-6\\9\end{pmatrix}, \quad \ell_2: \; \mathbf{r} = \begin{pmatrix}0\\1\\2\end{pmatrix} + s\begin{pmatrix}1\\-2\\3\end{pmatrix}$ are parallel but not coincident.

Find the value of $m$ for which the lines $\ell_1: \; \mathbf{r} = \begin{pmatrix}1\\0\\2\end{pmatrix} + t\begin{pmatrix}1\\m\\-1\end{pmatrix}, \quad \ell_2: \; \mathbf{r} = \begin{pmatrix}3\\-1\\1\end{pmatrix} + s\begin{pmatrix}2\\1\\0\end{pmatrix}$ intersect.

For what value(s) of $m$ are the 3D lines $\ell_1: \; \mathbf{r} = \begin{pmatrix}0\\1\\2\end{pmatrix} + t\begin{pmatrix}1\\1\\1\end{pmatrix}, \quad \ell_2: \; \mathbf{r} = \begin{pmatrix}2\\0\\-1\end{pmatrix} + s\begin{pmatrix}1\\-1\\m\end{pmatrix}$ skew?

Find the value of $m$ for which the lines $\ell_1: \; \mathbf{r} = \begin{pmatrix}1\\0\\1\end{pmatrix} + t\begin{pmatrix}1\\2\\3\end{pmatrix}, \quad \ell_2: \; \mathbf{r} = \begin{pmatrix}m\\1\\0\end{pmatrix} + s\begin{pmatrix}2\\-1\\4\end{pmatrix}$ are coplanar.

Find the value of $m$ such that the lines $\ell_1: \; \mathbf{r} = \begin{pmatrix}1\\ m\\0\end{pmatrix} + t\begin{pmatrix}1\\2\\-1\end{pmatrix}, \quad \ell_2: \; \mathbf{r} = \begin{pmatrix}0\\1\\1\end{pmatrix} + s\begin{pmatrix}2\\-1\\0\end{pmatrix}$ intersect at right angles (are perpendicular at their point of intersection).

Find all values of $m$ such that the acute angle between the lines $\ell_1: \; \mathbf{r} = \begin{pmatrix}0\\0\\0\end{pmatrix} + t\begin{pmatrix}1\\2\\-1\end{pmatrix}, \quad \ell_2: \; \mathbf{r} = \begin{pmatrix}1\\-1\\2\end{pmatrix} + s\begin{pmatrix}2\\ m\\1\end{pmatrix}$ is $60^\circ$.

Determine the value of $m$ for which the vectors $(1,1,m)$, $(3,1,2)$ and the connecting vector from $(2,3,0)$ to $(0,4,1)$ are coplanar.

Find all $m$ such that the shortest distance between the skew lines $\ell_1: \; \mathbf{r} = \begin{pmatrix}0\\0\\0\end{pmatrix} + t\begin{pmatrix}1\\1\\0\end{pmatrix}, \quad \ell_2: \; \mathbf{r} = \begin{pmatrix}0\\1\\m\end{pmatrix} + s\begin{pmatrix}0\\1\\1\end{pmatrix}$ is $\dfrac{2}{\sqrt{3}}$.