The equation of the plane passing through the line of intersection of the planes r → · 2 i ^ - 3 j ^ + 4 k ^ = 1 and r → · i ^ - j ^ + 4 = 0 and perpendicular to the plane r → · 2 i ^ - j ^ - k ^ + 4 = 0 , is

r → · i ^ - 2 j ^ + 4 k ^ = 5

r → · i ^ - 2 j ^ + 4 k ^ = 3

r → · 2 i ^ - j ^ + 5 k ^ = 3

r → · 2 i ^ - j ^ + 5 k ^ = 5

The vector equation of the plane passing through the intersection of the planes r → · i ^ + j ^ + k ^ = 1 and r → · i ^ - 2 j ^ = - 2 , and the point 1 , 0 , 2 is:

r → · i ^ + 7 j ^ + 3 k ^ = 7

r → · i ^ - 7 j ^ + 3 k ^ = 7 3

r → · i ^ + 7 j ^ + 3 k ^ = 7 3

r → · 3 i ^ + 7 j ^ + 3 k ^ = 7

The equation of the plane passing through the line of intersection of the planes r → · ( i ^ + j ^ + k ^ ) = 1 and r → · ( 2 i ^ + 3 j ^ - k ^ ) + 4 = 0 and parallel to the x -axis, is

r → · ( j ^ - 3 k ^ ) + 6 = 0

r → · ( i ^ + 3 k ^ ) + 6 = 0

r → · ( i ^ - 3 k ^ ) + 6 = 0

r → · ( j ^ - 3 k ^ ) - 6 = 0

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The plane $a x + b y + c z + d = 0$ bisects an angle between a pair of planes of which one is $l x + m y + n z + p = 0$ the equation of the other plane of pair is

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Important Questions on Three Dimensional Geometry

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

The equation of the plane passing through the line of intersection of the planes

\vec{r} \cdot (2 \hat{i} - 3 \hat{j} + 4 \hat{k}) = 1

and

\vec{r} \cdot (\hat{i} - \hat{j}) + 4 = 0

and perpendicular to the plane

\vec{r} \cdot (2 \hat{i} - \hat{j} - \hat{k}) + 4 = 0

, is

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

The equation of the plane containing the line of intersection of

2 x - 5 y + z = 3; x + y + 4 z = 5,

and parallel to the plane,

x + 3 y + 6 z = 1,

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

If the equation of the plane passing through the line of intersection of the planes

2 x - 7 y + 4 z - 3 = 0, 3 x - 5 y + 4 z + 11 = 0

and the point

(- 2, 1, 3)

a x + b y + c z - 7 = 0,

then the value of

2 a + b + c - 7

is _________.

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

Let the equation of the plane, that passes through the point

(1, 4, - 3)

and contains the line of intersection of the planes

3 x - 2 y + 4 z - 7 = 0

and

x + 5 y - 2 z + 9 = 0

, be

α x + β y + γ z + 3 = 0,

then

α + β + γ

is equal to :

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

The equation of the plane passing through the intersection of the planes

x + 2 y - z = 3

and

x + y - 3 z = 5

and passing through the point

(1, - 1, 0)

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

Equation of a plane at a distance

\sqrt{\frac{2}{21}}

units from the origin, which contains the line of intersection of the planes

x - y - z - 1 = 0

and

2 x + y - 3 z + 4 = 0,

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

The vector equation of the plane passing through the intersection of the planes

\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1

and

\vec{r} \cdot (\hat{i} - 2 \hat{j}) = - 2,

and the point

(1, 0, 2)

is:

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

The equation of the plane passing through the line of intersection of the planes

x + y + z = 6

and

2 x + 3 y + 4 z + 5 = 0

and passing through

(1, 1, 1)

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

If the equation of a plane

P

, passing through the intersection of the planes,

x + 4 y - z + 7 = 0

and

3 x + y + 5 z = 8

a x + b y + 6 z = 15

for some

a, b \in R,

then the distance of the point

(3, 2, - 1)

from the plane

P

\dots \dots

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

The equation of the plane through the intersection of the planes

x + y + z = 1

and

2 x + 3 y - z + 4 = 0

and parallel to

X

-axis is

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

The vector in the direction of the sum of the vectors

\vec{a} = 2 \hat{i} - 2 \hat{j} + 5 \hat{k} and \vec{b} = - 2 \hat{i} + 5 \hat{j} - 3 \hat{k}

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

The equation of a plane containing the line of intersection of the planes

2 x - y - 4 = 0

and

y + 2 z - 4 = 0

and passing through the point

(1,1, 0)

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

X

intercept of the plane containing the line of intersection of the planes

x - 2 y + z + 2 = 0

and

3 x - y - z + 1 = 0

and also passing through

(1, 1, 1)

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

A plane

P

contains the line

x + 2 y + 3 z + 1 = 0 = x - y - z - 6

, and is perpendicular to the plane

- 2 x + y + z + 8 = 0

. Then which of the following points lies on

P ?

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

The equation of the plane passing through the line of intersection of the planes

\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1

and

\vec{r} \cdot (2 \hat{i} + 3 \hat{j} - \hat{k}) + 4 = 0

and parallel to the

x

-axis, is

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

The equation of a plane passing through the line of intersection of the planes

x + 2 y + 3 z = 2

and

x - y + z = 3

and at a distance

\frac{2}{\sqrt{3}}

from the point

(3, 1, - 1)

is -

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

The vector equation of a plane passing through the line of intersection of the planes

\vec{r} \cdot \vec{n_{1}} = q_{1} and \vec{r} \cdot {\vec{n}}_{2} = q_{2} is

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

Determine the plane through the intersection of the planes

x + 2 y + 3 z - 4 = 0

and

2 x + y - z + 5 = 0

and perpendicular to the plane

5 x + 3 y + 6 z + 8 = 0

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

The vector equation of the plane through the line of intersection of the planes

x + y + z = 1

and

2 x + 3 y + 4 z = 5

which is perpendicular to the plane

x - y + z = 0

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Family of Planes

Let

P

be the plane, which contains the line of intersection of the planes,

x + y + z - 6 = 0

and

2 x + 3 y + z + 5 = 0

and it is perpendicular to the

x y

-plane. Then the distance of the point

(0, 0, 256)

from

P

is equal to:

The plane ax+by+cz+d=0 bisects an angle between a pair of planes of which one is lx+my+nz+p=0 the equation of the other plane of pair is

Important Questions on Three Dimensional Geometry

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The plane $a x + b y + c z + d = 0$ bisects an angle between a pair of planes of which one is $l x + m y + n z + p = 0$ the equation of the other plane of pair is