If the equation of a floor of a room is given by $x + y - z + 4 = 0$ and the equation of roof is given by $x + y - z + 5 = 0$. Then, the height of the room is :

The acute angle between the lines $\vec{r} = (4\hat{i} - \hat{j}) + \lambda(2\hat{i} + \hat{j} - 3\hat{k})$ and $\frac{x-1}{1} = \frac{y+1}{-3} = \frac{z-2}{2}$ is

Distance of the point $(2, 4, -1)$ from the line $\frac{10+2x}{2} = \frac{y+3}{4} = \frac{6-z}{9}$ is

Match List I with List II

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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?

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Let $L_1$ and $L_2$ be any two lines given by

$L_1: 3\hat{i}+2\hat{j}-4\hat{k} + \lambda(\hat{i}+2\hat{j}+2\hat{k})$

$L_2: 5\hat{i}-2\hat{j} + \mu(3\hat{i}+2\hat{j}+6\hat{k})$ Where $\lambda$ and $\mu$ are parameters

then which of the following statements are TRUE?

A. Equation of a line passing through the point $(1,2,3)$ which is parallel to line $L_1$ is $\dfrac{x-1}{1} = \dfrac{y-2}{2} = \dfrac{z-3}{2}$

B. Direction cosines of line $L_2$ are $\dfrac{3}{7}, \dfrac{2}{7}, \dfrac{6}{7}$

C. Line $L_1$ is parallel to the line $L_2$

D. The angle between the two lines $L_1$ and $L_2$ is $\cos^{-1}\left(\dfrac{19}{21}\right)$

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Match List-I with List-II

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