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The angle between the lines $\vec{r} = 3\hat{i} + 2\hat{j} - 4\hat{k} + \lambda(\hat{i} + 2\hat{j} + 2\hat{k})$ and $\vec{r} = 5\hat{j} - 2\hat{k} + \mu(3\hat{i} + 2\hat{j} + 6\hat{k})$ is :

The value of $\lambda$, so that the lines $\frac{1-x}{3} = \frac{7y-14}{2\lambda} = \frac{z-3}{2}$ and $\frac{7-7x}{3\lambda} = \frac{y-5}{1} = \frac{6-z}{5}$ are perpendicular is:

Match List I with List II

Choose the correct answer from the options given below:

Distance between the point (3, 4, 5) and the point where the line $\frac{x-3}{1} = \frac{y-4}{2} = \frac{z-5}{2}$ meets the plane $x + y + z = 17$ is :

Cartesian equation of plane passing through the points (2, -4, 5) and perpendicular to the line with direction ratios (3, -1, 2) is :

If the shortest distance between the lines $l_1$ and $l_2$ given by $\vec{r} = a\hat{i} + 2\hat{j} - \hat{k} + \lambda(2\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = \hat{i} - \hat{j} + \hat{k} + \mu(2\hat{i} - \hat{j} + \hat{k})$ is $\sqrt{\frac{35}{6}}$ units, the values of 'a' can be :

The vector equation of the line joining the points $(-2, -3, -4)$ and $(1, -2, 4)$ is :

The angle between the two planes $x + y - z = 3$ and $3x + 2y + z = 5$ is :

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 equation of plane which cuts equal intercepts of unit length on the coordinate axes is :

If the straight lines $x = 1 + s$, $y = -3 - \lambda s$, $z = 1 + \lambda s$ and $x = \frac{t}{2}$, $y = 1 + t$, $z = 2 - t$ with parameters $s$ and $t$ respectively, are coplanar, then $\lambda$ is equal to :

The shortest distance between the lines $\frac{x+3}{1} = \frac{y-2}{2} = \frac{z+4}{3}$ and $\frac{x+3}{-3} = \frac{y+7}{2} = \frac{z-6}{4}$ is :

Find the root of perpendicular from origin to the line $\frac{x - 1}{3} = \frac{y - 2}{-2} = \frac{z + 1}{3}$.

The distance of the point $(2, 3, -5)$ from the plane $x + 2y - 2z = 9$ is :

Equation of y-axis in space, in vector form is :

The angle between 2 planes $4x + 8y + z - 8 = 0$ and $y + z - 4 = 0$ is :

The equation of plane that contains line $\frac{x - 1}{-1} = \frac{y + 1}{2} = \frac{z - 1}{3}$ and also pass through point $(0, 1, 0)$ is :

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

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

A line makes angle $\theta$ with $x$-axes as well as $z$-axis. If the angle $\beta$, which it makes with $y$-axis is such that $\sin^2\beta = 3\sin^2\theta$, then $\cos^2\theta$ is :

The equation of the plane, parallel to the plane $3x + 4y - 12z = 3$ and passes through $(1, 1, -1)$, is :

The distance of plane $\vec{r}\cdot(6\hat{i} - 3\hat{j} - 2\hat{k}) + 1 = 0$ from origin is :

The foot of perpendicular from point $(2, 4, -1)$ on the line $\frac{x+5}{1} = \frac{y+3}{4} = \frac{z-6}{-9}$ is :

If two lines $\frac{x-3}{2} = \frac{y-4}{5} = \frac{z}{4}$ and $\frac{x-4}{3} = \frac{y-5}{6} = \frac{1-z}{k}$, are coplanar, then $k$ is equal to

The foot of perpendicular from the point P (1, 2, -3) to the line $\frac{x+1}{2} = \frac{y-3}{-2} = \frac{z}{-1}$ is

The equation of plane passing through the point of (3, 2, 0) and containing the line $\frac{x-2}{2} = \frac{y+3}{4} = \frac{z-1}{1}$ is

The sum of all the values of $\lambda$ for which the distance of the point P (2, 3, $\lambda$) from the plane $x + 2y - 2z = 9$ is 3 units, is

The points of trisection of the segment joining the points (1, 0, 2) and (1, 3, 2) are:

A. $(1, \frac{3}{2}, \frac{4}{3})$

B. (1, 1, 2)

C. (1, 2, 2)

D. $(1, \frac{3}{2}, 2)$

Choose the correct answer from the options given below:

The equation of line of intersection of the planes $x + y + 3z = 7$ and $x - y + 2z = 3$ is:

The distance of the point (1, 2, 0) from the line $\frac{x-3}{2} = \frac{y+4}{3} = \frac{z+6}{5}$ measured parallel to the plane $x + y + z = 3$ is

The line $\frac{x+2}{3} = \frac{y+3}{5} = \frac{z-6}{4}$ passes through (a, 2, c). The value of a and c are: