The line of intersection of the planes r → ⋅ 3 i ^ - j ^ + k ^ = 1 and r → ⋅ i ^ + 4 j ^ - 2 k ^ = 2 , is,

x - 6 13 2 = y - 5 13 - 7 = z - 13

x - 6 13 2 = y - 5 13 7 = z - 13

MEDIUM

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Let $σ_{1}, σ_{2}, σ_{3}$ be planes passing through the origin. Assume that $σ_{1}$ is perpendicular to the vector $(1, 1, 1)$ , $σ_{2}$ is perpendicular to a vector $(a, b, c)$ and $σ_{3}$ is perpendicular to the vector $(a^{2}, b^{2}, c^{2})$ . What are all the positive values of $a, b$ and $c$ so that $σ_{1} \cap σ_{2} \cap σ_{3}$ is a single point?

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Important Questions on Three Dimensional Geometry

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

The equation of plane containing line

x - y = 1, z = 1

and parallel to

\frac{x}{2} - \frac{z}{3} = 1, y = 3

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

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

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

The angle between the planes

x + y + z = 1

and

x - 2 y + 3 z = 1

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

Vectors

\vec{a}, \vec{b}, \vec{c}, \vec{d}

are such that

(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) = \vec{0} . P_{1}

and

P_{2}

are two planes determined by vectors

\bar{a}, \bar{b}

and

\bar{c}, \bar{d}

, respectively. Then the angle between the planes

P_{1}

and

P_{2}

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

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 Q R

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

An equation of plane parallel to plane

x - 2 y + 2 z - 5 = 0

and at a unit distance from the origin is

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

The equation of the plane passing through the point

(1, - 1, 2)

and parallel to the plane

3 x + 4 y - 5 z = 0

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

Let

L

be a line obtained from the intersection of two planes

x + 2 y + z = 6

and

y + 2 z = 4

. If point

P (α, β, γ)

is the foot of perpendicular from

(3, 2, 1)

L

, then the value of

21 (α + β + γ)

equals:

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

The distance of the point

(1, - 2, 4)

from the plane passing through the point

(1, 2, 2)

and perpendicular to the planes

x - y + 2 z = 3

and

2 x - 2 y + z + 12 = 0,

is :

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

The planes

2 x - y + 4 z = 5

and

5 x - 2.5 y + 10 z = 6

are

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

π_{1}

is a plane passing through the point

(1, 2, 3)

and perpendicular to the planes

x + 2 y + 3 z - 6 = 0, x + 2 y + 2 z - 5 = 0

. If

(- 1, 2, - 3)

is the foot of the perpendicular drawn from the point

(1, 3, 2)

on to a plane

π_{2},

then the angle between the planes

π_{1}

and

π_{2}

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

If the angle between the planes

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

and

\vec{r} . (2 \hat{i} - m \hat{j} - \hat{k}) - 5 = 0

\frac{π}{3}

then

m =

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

The locus represented by

x y + y z = 0

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

If the planes

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

and

\vec{r} \cdot (λ \hat{i} + 5 \hat{j} - \hat{k}) = 5

are perpendicular to each other, then the value of

λ^{2} + λ

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

Consider the three planes

$P_{1} : 3 x + 15 y + 21 z = 9$

$P_{2} : x - 3 y - z = 5,$ and

$P_{3} : 2 x + 10 y + 14 z = 5$

Then, which one of the following is true?

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

The equation

(x - 1) (x - 2) = 0

represents, in three dimensional space, a:

EASY

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

The equation of the planes parallel to the plane

x - 2 y + 2 z - 3 = 0

which are at unit distance from the point

(1, 2, 3)

a x + b y + c z + d = 0 .

(b - d) = K (c - a),

then the positive value of

K

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

If the plane

2 x - y + 2 z + 3 = 0

has the distances

\frac{1}{3}

and

\frac{2}{3}

units from the planes

4 x - 2 y + 4 z + λ = 0

and

2 x - y + 2 z + μ = 0

, respectively, then the maximum value of

λ + μ

is equal to:

HARD

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

The line of intersection of the planes

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

and

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

is,

MEDIUM

Mathematics>Vectors and 3D Geometry>Three Dimensional Geometry>Interaction between Two Planes

Distance between two parallel planes

2 x + y + 2 z = 8

and

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

Let σ1, σ2, σ3 be planes passing through the origin. Assume that σ1 is perpendicular to the vector 1,1,1, σ2 is perpendicular to a vector a,b,c and σ3 is perpendicular to the vector a2,b2,c2. What are all the positive values of a, b and c so that σ1∩σ2∩σ3 is a single point?

Important Questions on Three Dimensional Geometry

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