The shortest distance between the lines $\dfrac{x+1}{7}=\dfrac{-y-1}{6}=\dfrac{z+1}{1}$ and $\dfrac{x-3}{1}=\dfrac{-y+5}{2}=\dfrac{z-7}{1}$ is

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 $\cos^{-1}\left(\dfrac{19}{21}\right)$

Choose the correct answer from the options given below:

The angle at which the line, $\frac{x-1}{0} = \frac{2-y}{-1} = \frac{2z-3}{-2}$ is inclined with the positive direction of z-axis is

The direction cosines of the line which is perpendicular to the lines with direction ratios $1,-2,-2$ and $0,2,1$ are :

If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (−4, 3, −6) and (2, 9, 2) respectively, then the angle between the lines AB and CD is

Consider a line $\vec{r} = (\hat{i} + 4\hat{j}) + \lambda(2\hat{i} - 2\hat{j} + 3\hat{k})$, then which of the following statements are correct?

(A) it passes through point (9, -4, 12)

(B) it passes through point (1, 4, -1)

(C) its direction cosine's are $\frac{2}{\sqrt{17}}, \frac{-2}{\sqrt{17}}, \frac{3}{\sqrt{17}}$

(D) its Cartesian equation is $\frac{x - 1}{2} = \frac{y - 4}{-2} = \frac{z}{3}$

Choose the correct answer from the options given below:

The distance between the lines $\vec{r}=\hat{i}-2 \hat{j}+3 \hat{k}+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})$ and $\vec{r}=3 \hat{i}-2 \hat{j}+1 \hat{k}+\mu(4 \hat{i}+6 \hat{j}+12 \hat{k})$ is :