If the lines $\frac{x-1}{2}=\frac{y+2}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect,then $k$ has the value

Find the shortest distance between the lines $l_{1}$ and $l_{2}$ whose vector equations are
$\vec{r}=\hat{i}+\hat{j}+\lambda(2 \hat{i}-\hat{j}+\hat{k})$ $(1)$
and $\vec{r}=2 \hat{i}+\hat{j}-\hat{k}+\mu(3 \hat{i}-5 \hat{j}+2 \hat{k})$ $(2)$

The vector equation of the line whose Cartesian equations are $y=2$ and $4x-3z+5=0$ is

If the lines with direction cosines $(l, m, n)$ satisfying $al + bm + cn = 0$ and $fmn + gnl + hlm = 0$ are perpendicular,then $\frac{f}{a} + \frac{g}{b} + \frac{h}{c} = .........$

If the two lines $\frac{x - 1}{2} = \frac{y - 1}{3} = \frac{z - 1}{4}$ and $\frac{x - 3}{1} = \frac{y - k}{2} = \frac{z}{1}$ intersect at a point,then find the value of $k$.

The perpendicular distance from the point $P(-1, 1, 0)$ to the line joining the points $A(0, 2, 4)$ and $B(3, 0, 1)$ is