The shortest distance between the skew lines $r = (-\hat{i} + 3\hat{k}) + t(2\hat{i} + 3\hat{j} + 6\hat{k})$ and $r = (3\hat{i} + \hat{j} - \hat{k}) + s(2\hat{i} - \hat{j} + 2\hat{k})$ 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 .......

The cosine of the angle included between the lines $\overline{r}=(2 \hat{\imath}+\hat{\jmath}-2 \hat{k})+\lambda(\hat{\imath}-2 \hat{\jmath}-2 \hat{k})$ and $\overline{r}=(\hat{\imath}+\hat{\jmath}+3 \hat{k})+\mu(3 \hat{\imath}+2 \hat{\jmath}-6 \hat{k})$ where $\lambda, \mu \in R$ is

The point $P$ lies on the line $AB$,where $A \equiv (2, 4, 5)$ and $B \equiv (1, 2, 3)$. If the $z$-coordinate of point $P$ is $3$,then its $y$-coordinate is:

The vector equation of the line passing through $P(1, 2, 3)$ and $Q(2, 3, 4)$ is

The shortest distance between the lines $\frac{x - 3}{2} = \frac{y + 15}{-7} = \frac{z - 9}{5}$ and $\frac{x + 1}{2} = \frac{y - 1}{1} = \frac{z - 9}{-3}$ is