Let $(\alpha, \beta, \gamma)$ be the mirror image of the point $(2, 3, 5)$ in the line $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$. Then $2\alpha + 3\beta + 4\gamma$ is equal to

Let $P$ be the point $(10, -2, -1)$ and $Q$ be the foot of the perpendicular drawn from the point $R(1, 7, 6)$ on the line passing through the points $(2, -5, 11)$ and $(-6, 7, -5)$. Then the length of the line segment $PQ$ is equal to ..........

The point of intersection of the lines represented by $r=(\hat{i}+2 \hat{j}-\hat{k})+\lambda(2 \hat{i}+3 \hat{j}+4 \hat{k})$ and $r=(-\hat{i}-3 \hat{j}+7 \hat{k})+\mu(\hat{i}+2 \hat{j}-\hat{k})$ is

The direction ratios of a bisector of the angle between two lines whose direction ratios are $1, 1, 2$ and $\sqrt{3}, -\sqrt{3}, 0$ are

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 Cartesian equation of the line which is parallel to the line $\frac{x+3}{3}=\frac{y-4}{5}=\frac{z+8}{6}$ and passing through the point $(1, -3, 5)$ is: