If the line joining the points $A(2, 3, -1)$ and $B(3, 5, -3)$ is perpendicular to the line joining the points $C(1, 2, 3)$ and $D(3, y, 7)$,then $y=$

If the shortest distance between the lines
$L_1: \overrightarrow{r}=(2+\lambda) \hat{i}+(1-3 \lambda) \hat{j}+(3+4 \lambda) \hat{k}, \lambda \in R$
$L_2: \overrightarrow{r}=2(1+\mu) \hat{i}+3(1+\mu) \hat{j}+(5+\mu) \hat{k}, \mu \in R$
is $\frac{m}{\sqrt{n}}$,where $\operatorname{gcd}(m, n)=1$,then the value of $m+n$ equals.

If the line joining points $(2, 1, 4)$ and $(a - 1, 4, -1)$ is parallel to the line joining points $(0, 2, b - 1)$ and $(5, 3, -2)$, then the values of $a$ and $b$ are respectively:

If lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z-0}{1}$ intersect,then the value of $k$ is

The angle between the lines $\bar{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda(\hat{i}+\hat{j}+2\hat{k})$ and $\bar{r}=(3\hat{i}+\hat{k})+\lambda^{\prime}(2\hat{i}+\hat{j}-\hat{k})$,where $\lambda, \lambda^{\prime} \in R$,is

Show that the points $(2,3,4), (-1,-2,1),$ and $(5,8,7)$ are collinear.