Angle Between Two Planes

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JEE Main 2003

A tetrahedron has vertices at $O (0, 0, 0), A (1, 2, 1), B (2, 1, 3)$ and $C (- 1, 1, 2)$ . Then the angle between the faces OAB and ABC will be

9 0^{\circ}

(B)

cos^{- 1} (\frac{19}{35})

(C)

cos^{- 1} (\frac{17}{31})

(D)

3 0^{\circ}

JEE Main 2002

LEVELJEE Main

The d.r. of normal to the plane through $(1, 0, 0), (0, 1, 0)$ which makes an angle $π /4$ with plane $x + y = 3$ are

(A)

1, 2, 1

(B)

1, 1, 2

(C)

1, 1, 2

(D)

2, 1, 1

JEE Main 2019 (12 January Shift 1)

LEVELJEE Advanced

A tetrahedron has vertices $P (1, 2, 1)$ , $Q (2, 1, 3)$ , $R (- 1, 1, 2)$ and $O (0, 0, 0)$ . The angle between the faces $O P Q$ and $P QR$ is :

(A)

cos^{- 1} (9/35)

(B)

cos^{- 1} (19/35)

(C)

cos^{- 1} (17/31)

(D)

cos^{- 1} (7/31)

JEE Main 2019 (12 April)

LEVELJEE Main

A plane which bisects the angle between the two given planes $2 x - y + 2 z - 4 = 0$ and $x + 2 y + 2 z - 2 = 0$ , passes through the point:

JEE Main 2021 (01 Sep Shift 2)

LEVELJEE Main

Let the acute angle bisector of the two planes $x - 2 y - 2 z + 1 = 0$ and $2 x - 3 y - 6 z + 1 = 0$ be the plane $P$ . Then which of the following points lies on $P$ ?

(A)

(3, 1, - \frac{1}{2})

(B)

(- 2, 0, - \frac{1}{2})

(C)

(0, 2, - 4)

(D)

(4, 0, - 2)

JEE Main 2022 (24 June Shift 2)

LEVELJEE Main

Let the points on the plane $P$ be equidistant from the points $(- 4, 2, 1)$ and $(2, - 2, 3)$ . Then the acute angle between the plane $P$ and the plane $2 x + y + 3 z = 1$ is

(A)

\frac{π}{6}

(B)

\frac{π}{4}

(C)

\frac{π}{3}

(D)

\frac{5 π}{12}

JEE Main 2022 (28 June Shift 1)

LEVELJEE Main

The acute angle between the planes $P_{1}$ and $P_{2}$ , when $P_{1}$ and $P_{2}$ are the planes passing through the intersection of the planes $5 x + 8 y + 13 z - 29 = 0$ and $8 x - 7 y + z - 20 = 0$ and the points $(2, 1, 3)$ and $(0, 1, 2)$ , respectively, is

(A)

\frac{π}{3}

(B)

\frac{π}{4}

(C)

\frac{π}{6}

(D)

\frac{π}{12}

JEE Main 2023 (30 January Shift 1)

LEVELJEE Advanced

If $λ_{1} < λ_{2}$ are two values of $λ$ such that the angle between the planes $P_{1} : r \cdot (3 \hat{i} - 5 \hat{j} + \hat{k}) = 7$ and $P_{2} : r \cdot (λ \hat{i} + \hat{j} - 3 \hat{k}) = 9$ is $sin^{- 1} (\frac{2 6}{5})$ , then the square of the length of perpendicular from the point $(38 λ_{1}, 10 λ_{2}, 2)$ to the plane $P_{1}$ is ______.

JEE Main 2023 (31 January Shift 1)

LEVELJEE Main

Let $θ$ be the angle between the planes $P_{1} = r \cdot (\hat{i} + \hat{j} + 2 \hat{k}) = 9$ and $P_{2} = r \cdot (2 \hat{i} - \hat{j} + \hat{k}) = 15$ . Let $L$ be the line that meets $P_{2}$ at the point $(4, - 2, 5)$ and makes an angle $θ$ with the normal of $P_{2}$ . If $α$ is the angle between $L$ and $P_{2}$ then $(tan^{2} θ) (cot^{2} α)$ is equal to ______.

JEE(ADVANCED)-201

LEVELJEE Advanced

Let $P_{1} : 2 x + y - z = 3$ and $P_{2} : x + 2 y + z = 2$ be two planes. Then, which of the following statement(s) is (are) TRUE ?

* Multiple Correct Options

(A)

The line of intersection of

P_{1}

and

P_{2}

has direction ratios 1, 2, -1

(B)

The line

\frac{3 x - 4}{9} = \frac{1 - 3 y}{9} = \frac{z}{3}

is perpendicular to the line of intersection of

P_{1}

and

P_{2}

(C)

The acute angle between

P_{1}

and

P_{2}

6 0^{\circ}

(D)

P_{3}

is the plane passing through the point (4, 2, -2) and perpendicular to the line of intersection of

P_{1}

and

P_{2}

, then the distance of the point (2, 1, 1) from the plane

P_{3}

\frac{2}{3}

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A tetrahedron has vertices at O(0,0,0),A(1,2,1),B(2,1,3) and C(−1,1,2). Then the angle between the faces OAB and ABC will be

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The d.r. of normal to the plane through (1,0,0),(0,1,0) which makes an angle π/4 with plane x+y=3 are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A tetrahedron has vertices P(1,2,1), Q(2,1,3), R(−1,1,2) and O(0,0,0). The angle between the faces OPQ and PQR is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A plane which bisects the angle between the two given planes 2x−y+2z−4=0 and x+2y+2z−2=0, passes through the point:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the acute angle bisector of the two planes x−2y−2z+1=0 and 2x−3y−6z+1=0 be the plane P. Then which of the following points lies on P?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the points on the plane P be equidistant from the points (−4,2,1) and (2,−2,3). Then the acute angle between the plane P and the plane 2x+y+3z=1 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P1​:2x+y−z=3 and P2​:x+2y+z=2 be two planes. Then, which of the following statement(s) is (are) TRUE ?

A tetrahedron has vertices at $O (0, 0, 0), A (1, 2, 1), B (2, 1, 3)$ and $C (- 1, 1, 2)$ . Then the angle between the faces OAB and ABC will be

The d.r. of normal to the plane through $(1, 0, 0), (0, 1, 0)$ which makes an angle $π /4$ with plane $x + y = 3$ are

A tetrahedron has vertices $P (1, 2, 1)$ , $Q (2, 1, 3)$ , $R (- 1, 1, 2)$ and $O (0, 0, 0)$ . The angle between the faces $O P Q$ and $P QR$ is :

A plane which bisects the angle between the two given planes $2 x - y + 2 z - 4 = 0$ and $x + 2 y + 2 z - 2 = 0$ , passes through the point:

Let the acute angle bisector of the two planes $x - 2 y - 2 z + 1 = 0$ and $2 x - 3 y - 6 z + 1 = 0$ be the plane $P$ . Then which of the following points lies on $P$ ?

Let the points on the plane $P$ be equidistant from the points $(- 4, 2, 1)$ and $(2, - 2, 3)$ . Then the acute angle between the plane $P$ and the plane $2 x + y + 3 z = 1$ is

Let $P_{1} : 2 x + y - z = 3$ and $P_{2} : x + 2 y + z = 2$ be two planes. Then, which of the following statement(s) is (are) TRUE ?