If a line makes angles $\alpha, \beta, \gamma$ with the positive directions of $x$ - axis, y -axis, z -axis respectively, then the value of $cos2\alpha + cos2\beta + cos2\gamma$ is equal to

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

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

Which of the following statements are true?

(A) The vector equation of the line through the point (5, 2, -4) and parallel to the vector $3\hat{i} + 2\hat{j} - 8\hat{k}$ is $\vec{r} = (5\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(3\hat{i} + 2\hat{j} - 8\hat{k})$

(B) Vector form of the equation of line $\frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2}$ is $\vec{r} = (5\hat{i} - 4\hat{j} + 6\hat{k}) + \lambda(3\hat{i} + 7\hat{j} + 2\hat{k})$

(C) The direction cosines of z-axis are (1, 1,0).

(D) If a line has direction ratios 2, -1, -2, then its direction cosines are -2/3, -1/3, -2/3.

Choose the correct answer from the options given below:

If shortest distance between the lines $\vec{r} = \hat{i} - \hat{j} + \lambda(2\hat{i}+\hat{k})$ and

$\vec{r} = 2\hat{i} - \hat{j} + \mu(\hat{i}-\hat{j}-\hat{k})$ is $\dfrac{\sqrt{14}}{14}m$ units, the value of $m$ is

A line in space is uniquely determined if

(A) It passes through a given point and has a given direction.

(B) It passes through two given points.

(C) It passes through a given point.

(D) It has a given direction.

Choose the correct answer from the options given below:

Equation of a line passing through (1, 2, -3) which is parallel to the line $\frac{x+2}{3} = \frac{y-3}{-4} = \frac{z+5}{5}$ is