If A a → ; B b → ; C c → and D d → are four points such that a → = - 2 i ^ + 4 j ^ + 3 k ^ ; b → = 2 i ^ - 8 j ^ ; c → = i ^ - 3 j ^ + 5 k ^ ; d → = 4 i ^ + j ^ - 7 k ^ , d is the shortest distance between the lines A B and C D , then

d = A B → C D → A C → A B → × C D →

d = 0 , hence A B and C D intersect

If the angle between the planes given by 6 x 2 + 4 y 2 - 10 z 2 + 3 y z + 4 z x - 11 x y = 0 is cos - 1 k , then the value of ' k ' is equal to

We have, 6 x 2 + 4 y 2 - 10 z 2 + 3 y z + 4 z x - 11 x y = 0 If θ is the angle between two planes a 1 x + b 1 y + c 1 z + d 1 = 0 and a 2 x + b 2 y + c 2 z + d 2 = 0 , then cos θ = a 1 a 2 + b 1 b 2 + c 1 c 2 a 1 2 + b 1 2 + c 1 2 a 2 2 + b 2 2 + c 2 2 Here, a 1 a 2 + b 1 b 2 + c 1 c 2 = 6 + 4 - 10 = 0 , therefore cos θ = 0 ⇒ θ = π 2 = cos - 1 0 ∴ k = 0

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IMPORTANT

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Read the following statement carefully and identify the true statement-

$(a)$ Two lines parallel to a third line are parallel.

$(b)$ Two lines perpendicular to a third line are parallel.

$(c)$ Two lines parallel to a plane are parallel.

$(d)$ Two lines perpendicular to a plane are parallel.

$(e)$ Two lines either intersect or are parallel.

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Important Questions on Three Dimensional Geometry

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IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 22 - Three Dimensional Geometry>EXERCISE-2>Q 21

A parallelopiped is formed by planes drawn through the points

(1, 2, 3)

and

(9, 8, 5)

parallel to the coordinate planes then which of the following is the length of an edge of this rectangular parallelopiped-

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IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 22 - Three Dimensional Geometry>EXERCISE-2>Q 22

A (\vec{a}); B (\vec{b}); C (\vec{c})

and

D (\vec{d})

are four points such that

\vec{a} = - 2 \hat{i} + 4 \hat{j} + 3 \hat{k}; \vec{b} = 2 \hat{i} - 8 \hat{j}; \vec{c} = \hat{i} - 3 \hat{j} + 5 \hat{k}; \vec{d} = 4 \hat{i} + \hat{j} - 7 \hat{k}, d

is the shortest distance between the lines

A B

and

C D

, then

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IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 22 - Three Dimensional Geometry>EXERCISE-3>Q 17

The equation of the plane which has the property that the point

Q (5, 4, 5)

is the reflection of point

P (1, 2, 3)

through that plane, is

a x + b y + c z = d

where

a, b, c, d \in N

. Find the least value of

a + b + c + d

MEDIUM

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IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 22 - Three Dimensional Geometry>EXERCISE-3>Q 18

If the angle between the planes given by

6 x^{2} + 4 y^{2} - 10 z^{2} + 3 y z + 4 z x - 11 x y = 0

\cos^{- 1} (k)

, then the value of '

k

' is equal to

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IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 22 - Three Dimensional Geometry>EXERCISE-3>Q 19

Let the equation of the plane containing the line

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

and is parallel to the line of intersection of the planes

2 x + 3 y + z = 1

and

x + 3 y + 2 z = 2

x + A y + B z + C = 0

Compute the value of

(\frac{A + B + C}{2})

HARD

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IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 22 - Three Dimensional Geometry>EXERCISE-3>Q 20

Consider the plane

E : \vec{r} = [\begin{array}{r} - 1 \\ 1 \\ 1 \end{array}] + λ [\begin{array}{l} 1 \\ 2 \\ 0 \end{array}] + μ [\begin{array}{l} 1 \\ 0 \\ 1 \end{array}]

F

is a plane containing the point

A (- 4, 2, 2)

and parallel to

E

. Suppose the point

B

is on the plane

E

, such that

B

has a minimum distance from point

A

. If

C (- 3, 0, 4)

lies in the plane

F

. Then find the area of

Δ A B C

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IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 22 - Three Dimensional Geometry>EXERCISE-3>Q 21

Through a point

P (f, g, h)

, a plane is drawn at right angles to

O P

where '

O

' is the origin, to meet the coordinate axes in

A, B, C

. If the area of the triangle

A B C

\frac{r^{5}}{λ f g h}

where

O P = r

, then the value of

λ

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IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 22 - Three Dimensional Geometry>EXERCISE-3>Q 22

The position vectors of the four angular points of a tetrahedron

O A B C

are

(0, 0, 0); (0, 0, 2); (0, 4, 0)

and

(6, 0, 0)

respectively. A point

P

inside the tetrahedron is at the same distance '

r

' from the four plane faces of the tetrahedron. The value of

(\frac{3 r}{8})

Important Questions on Three Dimensional Geometry

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