One line is parameterized by
\[\begin{pmatrix} -1 + s \\ 3 - ks \\ 1 + ks \end{pmatrix}.\]Another line is parameterized by
\[\begin{pmatrix} t/2 \\ 1 + t \\ 2 - t \end{pmatrix}.\]If the lines are coplanar (i.e. there is a plane that contains both lines), then find $k.$
I know that two lines are coplanar if and only if they are parallel or intersecting. To check if they are parallel, I can compare their direction vectors, which are the coefficients of the parameters $s$ and $t$. The direction vector of the first line is $\begin{pmatrix} 1 \\ -k \\ k \end{pmatrix}$ and the direction vector of the second line is $\begin{pmatrix} 1/2 \\ 1 \\ -1 \end{pmatrix}.$ Two vectors are parallel if they are scalar multiples of each other, that is, if there is some constant $c$ such that $\begin{pmatrix} 1 \\ -k \\ k \end{pmatrix} = c \begin{pmatrix} 1/2 \\ 1 \\ -1 \end{pmatrix}.$ Comparing the first coordinates, I get $c = 2.$ Comparing the second coordinates, I get $-k = 2.$ Comparing the third coordinates, I get $k = -2.$ These equations are consistent, so the lines are parallel and $k = -2.$ 