The direction ratio's of line of intersection of two planes :

$2x + y + z + 47 = 0$ and $3x - 2y - z + 41 = 0$ are :

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:

The angle between the lines $\dfrac{3-x}{-7} = \dfrac{y+5}{-5} = \dfrac{z}{1}$ and $\dfrac{x}{1} = \dfrac{y}{2} = \dfrac{z}{3}$ is:

The shortest distance between the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x-2}{4} = \frac{y-4}{6} = \frac{z-5}{8}$ is equal to

If the direction ratios of two lines are $a, b, c$ and $(b-c), (c-a), (a-b)$ respectively, then the angle between these lines is:

Given a line $L:\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$, then

Match the LIST-I with LIST-II

Choose the correct answer from the options given below:

The two lines given by $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \mu(\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = (2\hat{i} - \hat{j} - \hat{k}) + \mu(-\hat{i} + \hat{j} - \hat{k})$

A. are perpendicular

B. are parallel.

C. have shortest distance 0.

D. have shortest distance $\sqrt{26}$.

E. have shortest distance $\sqrt{78}$.

Which of the above statements are true?

Choose the correct answer from the options given below: