JEE Main · Mathematics

Three dimensional geometry

20 practice questions, each with the answer and an explanation.

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Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

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What is the shortest distance between the two skew lines $r_{1} = (\hat{i} + 2 \hat{j} + 3 \hat{k}) + λ (\hat{i} - 3 \hat{j} + 2 \hat{k})$ and…
What is the perpendicular distance of the point $P (1, 2, 3)$ from the plane $2 x - 3 y + 6 z + 7 = 0$ ?
If the direction cosines of a directed straight line are $l, m, n$ , which of the following identities is identically true?
What is the angle $θ$ between the straight line $\frac{x - 1}{2} = \frac{y + 2}{1} = \frac{z - 3}{2}$ and the plane $2 x - y + 2 z - 5 = 0$ ?
Two lines $\frac{x - 1}{k} = \frac{y - 2}{2} = \frac{z - 3}{3}$ and $\frac{x - 2}{3} = \frac{y - 3}{k} = \frac{z - 1}{2}$ are perpendicular to each other. What is the value of $k$ ?
What is the equation of the plane passing through the point $(1, - 1, 2)$ and parallel to the plane $2 x - 3 y + 4 z = 6$ ?
The lines $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ and $\frac{x - 2}{3} = \frac{y - 3}{4} = \frac{z - 4}{5}$ are coplanar. What is the equation of the plane containing both lines?
What is the distance between the two parallel planes $2 x + y + 2 z = 8$ and $4 x + 2 y + 4 z + 5 = 0$ ?
Find the coordinates of the foot of the perpendicular drawn from the origin $(0, 0, 0)$ to the plane $2 x - 3 y + 4 z - 29 = 0$ .
What is the reflection (image) of the point $P (1, 3, 4)$ in the plane $2 x - y + z + 3 = 0$ ?
The angle between two lines whose direction ratios are $(1, 2, 2)$ and $(2, - 2, 1)$ is equal to:
A plane intercepts the coordinate axes at $(a, 0, 0)$ , $(0, b, 0)$ , and $(0, 0, c)$ such that the centroid of the triangle formed by these intercepts is $(1, 2, 3)$ . What is the equation of the plane?
What is the vector equation of the line passing through the point $(2, - 1, 4)$ and parallel to the vector $\hat{i} + 2 \hat{j} - \hat{k}$ ?
What is the Cartesian form of the line of intersection of the two planes $x + y + z = 1$ and $2 x - y + z = 2$ ?
The distance of the point $(2, 3, 4)$ from the plane $3 x - 6 y + 2 z + 11 = 0$ is:
A line makes angles $α, β, γ$ with the positive $x, y, z$ axes. What is the value of $sin^{2} α + sin^{2} β + sin^{2} γ$ ?
What is the equation of the plane passing through the three non-collinear points $(1, 1, 0)$ , $(1, 2, 1)$ , and $(- 2, 2, - 1)$ ?
The acute angle between the two planes $x + y + 2 z = 9$ and $2 x - y + z = 15$ is:
If the plane $2 x - 3 y + 6 z = k$ is tangent to the sphere $x^{2} + y^{2} + z^{2} = 4$ , what are the possible values of the constant $k$ ?
What is the shortest distance between the $z$ -axis and the line $x - y + 2 = 0, 2 x + z - 4 = 0$ ?

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Three dimensional geometry

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