If the lines $\frac{1-x}{2}=\frac{2y}{p}=\frac{z-1}{1}$ and $\frac{3-2x}{p}=\frac{y-1}{2}=\frac{z}{1}$ are parallel then value of 'p' is

Image of origin with respect to plane $x + y + z = 3$ is:

The shortest distance between the following lines:

$\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + s(2\hat{i} + \hat{j} + \hat{k})$

$\vec{r} = (\hat{i} + \hat{j} + 2\hat{k}) + t(4\hat{i} + 2\hat{j} + 2\hat{k})$, where s and t are scalars, is:

If the two lines given by

$x=py+q, z=ry+s$ and

$x=p'y+q', z=r'y+s'$

are perpendicular, then which of the following is equal to -1 ?

If $\theta$ is the angle between the lines $L_1$ and $L_2$, whose direction cosines are $l_1, m_1, n_1$ and $l_2, m_2, n_2$ respectively, then

Match List-I with List-II

Choose the correct answer from the options given below:

The shortest distance between the lines $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})$ and $\vec{r} = (2\hat{i} + 4\hat{j} + 5\hat{k}) + \mu(4\hat{i} + 6\hat{j} + 8\hat{k})$ is equal to

The angle between the pair of lines given by $\vec{r} = \hat{i} + 2\hat{j} - 3\hat{k} + \lambda (\hat{i} - 2\hat{j} + 2\hat{k})$ and $\vec{r} = 5\hat{i} + \hat{j} + \hat{k} + \mu (3\hat{i} - 2\hat{j} + 6\hat{k})$ is