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 each other,then the value of $k$ is:

The length of the perpendicular drawn from the origin to the line $\bar{r} = (4\hat{i} + 2\hat{j} + 4\hat{k}) + \lambda(3\hat{i} + 4\hat{j} - 5\hat{k})$ is .......

Let the line of the shortest distance between the lines $L_1: \vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k})$ and $L_2: \vec{r}=(4 \hat{i}+5 \hat{j}+6 \hat{k})+\mu(\hat{i}+\hat{j}-\hat{k})$ intersect $L_1$ and $L_2$ at $P$ and $Q$ respectively. If $(\alpha, \beta, \gamma)$ is the midpoint of the line segment $PQ$,then $2(\alpha+\beta+\gamma)$ is equal to . . . . . . .

The distance of the point $Q(0, 2, -2)$ from the line passing through the point $P(5, -4, 3)$ and perpendicular to the lines $\overrightarrow{r} = (-3 \hat{i} + 2 \hat{k}) + \lambda(2 \hat{i} + 3 \hat{j} + 5 \hat{k}), \lambda \in R$ and $\overrightarrow{r} = (\hat{i} - 2 \hat{j} + \hat{k}) + \mu(-\hat{i} + 3 \hat{j} + 2 \hat{k}), \mu \in R$ is

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

If the line passing through the points $(5, 1, a)$ and $(3, b, 1)$ crosses the $YZ$ plane at the point $(0, 17/2, -13/2)$,then $a+b=$