List of top Mathematics Questions on Three Dimensional Geometry

If $θ1,θ2, θ3$ are the angles made by a line with the positive directions of the $x,y,z$ axes, then the value of $\cos 2\theta_1 + \cos 2\theta_2 + \cos 2\theta_3$ is

Given the points $A(6,-7,0),B(16,-19,-4),C(0,3,-6)$ and $D(2,-5,10)$,the point of intersection of the lines $AB$ and $CD$ is

The shortest distance between the lines \[ \frac{x - 3}{2} = \frac{y - 2}{3} = \frac{z - 1}{-1} \] and \[ \frac{x + 3}{2} = \frac{y - 6}{1} = \frac{z - 5}{3} \] is:

What is the angle between the two straight lines $ y = (2 - \sqrt{3})x + 5 $ and $ y = (2 + \sqrt{3})x - 7 $?

If (2, 3, -1) is the foot of the perpendicular from (4, 2, 1) to a plane, then the equation of the plane is

The point of intersection of the line x + 1 = $\frac{y+3}{3}=\frac{-z+2}{2}$ with the plane 3x + 4y + 5z = 10 is

The length of perpendicular drawn from the point (3, -1, 11) to the line $\frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ is

If a lines makes an angle of $\frac{\pi}{3}$ with each X and Y axis then the acute angle made by Z-axis is

Let $\theta$ be the angle between the planes $P_1: \vec{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})=9$ and $P_2: \hat{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=15$ Let $L$ be the line that meets $P_2$ at the point $(4,-2,5)$ and makes an angle $\theta$ with the normal of $P_4$ If $\alpha$ is the angle between $L$ and $P_2$, then $\left(\tan ^2 \theta\right)\left(\cot ^2 \alpha\right)$ is equal to

One vertex of a rectangular parallelepiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing through O or P is :

Let N be the foot of perpendicular from the point P (1, –2, 3) on the line passing through the points (4, 5, 8) and (1, –7, 5). Then the distance of N from the plane 2x – 2y + z + 5 = 0 is

The plane, passing through the points (0, -1, 2) and (-1, 2, 1) and parellel to the line passing through (5,1,-7) and (1,-1,-1), also passes through the point

The line, that is coplanar to the line $\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}$ is

The distance of the point $P (4,6,-2)$ from the line passing through the point $(-3,2,3)$ and parallel to a line with direction ratios $3,3,-1$ is equal to :

Let the plane containing the line of intersection of the planes $P 1: x+(\lambda+4) y+z=1$ and $P 2: 2 x+$ $y+z=2$ pass through the points $(0,1,0)$ and $(1,0,1)$ Then the distance of the point $(2 \lambda, \lambda,-\lambda)$ from the plane $P 2$ is

Let $ \lambda_1 < \lambda_2 $ be two values of $ \lambda $ such that the angle between the planes:
$ P_1 : \vec{r} \cdot (3\hat{i} - 5\hat{j} + \hat{k}) = 7 $
$ P_2 : \vec{r} \cdot (\lambda\hat{i} + \hat{j} - 3\hat{k}) = 9 $
is $ \sin^{-1}\left(\frac{2\sqrt{6}}{5}\right) $. Then, the square of the length of the perpendicular from the point $ (38\lambda, 10\lambda, 2) $ to the plane $ P_1 $ is:

The distance of the point $(-1,9,-16)$ from the plane $2 x+3 y-z=5$ measured parallel to the line $\frac{x+4}{3}=\frac{2-y}{4}=\frac{z-3}{12}$ is

The perimeter of a △ABC is 6 times the arithmetic mean of the values of the sine of its angles. If the side BC is of the unit length, then ∠A =

Plane P₃ is passing through (1,1,1) and line of intersection of P₁ and P₂ where $P_{1}: 2x - y + z = 5$ and $P_{2}: x + 3y + 2z + 2 = 0$. Then distance of (1,1,10) from P₃ is:

If the image of point P(1, 2, 3) about the plane 2x - y + 3z = 2 is Q, then the area of triangle PQR, where coordinates of R is (4, 10, 12)

Let the line / : x = $\frac{1-y}{2}=\frac{z-3}{\lambda}, \lambda \in R$ meet the plane P : x+2y +3z = 4 at the point $(\alpha, \beta, \lambda)$. If the angle between the line I and the plane P is $cos^{-1}\bigg(\sqrt{\frac{5}{14}}\bigg)$ then $\alpha+2\beta+6\lambda$ is equal to______.

The point of intersection $C$ of the plane $8 x+y+2 z=0$ and the line joining the points $A (-3,-6,1)$ and $B (2,4,-3)$divides the line segment $AB$ internally in the ratio$k : 1 \ If a , b , c (| a |,| b |, | c |$are coprime) are the direction ratios of the perpendicular from the point $C$on the line $\frac{1-x}{1}=\frac{y+4}{2}=\frac{z+2}{3}$, then $| a + b + c |$is equal to ___