Find the shortest distance between the lines $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(2\hat{i} - \hat{j} - \hat{k})$ and $\vec{r} = (\hat{i} - \hat{j} - \hat{k}) + \mu(\hat{i})$.

The line passing through the points $(a, 1, 6)$ and $(3, 4, b)$ crosses the $yz$-plane at $\left(0, \frac{17}{2}, \frac{-13}{2}\right)$. Then the value of $(3a + 4b)$ is:

The vector equation of the line passing through the point $(1, 2, -4)$ and perpendicular to the two lines $\frac{x-8}{3} = \frac{y+19}{-16} = \frac{z-10}{7}$ and $\frac{x-15}{3} = \frac{y-29}{8} = \frac{z-5}{-5}$ is . . . . . . .

If a point $R(4, y, z)$ lies on the line segment joining the points $P(2, -3, 4)$ and $Q(8, 0, 10)$,then the distance of $R$ from the origin is

If the shortest distance between the lines $\overline{r}_1 = \alpha \hat{i} + 2 \hat{j} + 2 \hat{k} + \lambda(\hat{i} - 2 \hat{j} + 2 \hat{k})$ and $\overline{r}_2 = -4 \hat{i} - \hat{k} + \mu(3 \hat{i} - 2 \hat{j} - 2 \hat{k})$ is $9$,where $\lambda, \mu \in R$ and $\alpha > 0$,then the value of $\alpha$ is:

The point collinear with $(1, -2, -3)$ and $(2, 0, 0)$ among the following is