A = l 1 m 1 n 1 l 2 m 2 n 2 l 3 m 3 n 3 and B = p 1 q 1 r 1 p 2 q 2 r 2 p 3 q 3 r 3 p i , q i , r i are the cofactors of the elements l i , m i , n i for i = 1 ,2 , 3 . If l 1 , m 1 , n 1 , l 2 , m 2 , n 2 and l 3 , m 3 , n 3 are the direction cosines of three mutually perpendicular lines, then p 1 , q 1 , r 1 , p 2 , q 2 , r 2 and p 3 , q 3 , r 3 are

the direction ratios of three mutually perpendicular lines which are not direction cosines

the direction cosines of three lines which need not be perpendicular

the direction ratios but not the direction cosines of three lines which need not be perpendicular

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The shortest distance between the two lines $L_{1} : x = k_{1}; y = k_{2} & L_{2} : x = k_{3}; y = k_{4}$ is equal to

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Important Questions on Three Dimensional Coordinate System

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JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 3 - Three Dimensional Coordinate System>Three Dimensional Coordinate System Exercise 1>Q 92

A = [\begin{matrix} l_{1} & m_{1} & n_{1} \\ l_{2} & m_{2} & n_{2} \\ l_{3} & m_{3} & n_{3} \end{matrix}]

and

B = [\begin{matrix} p_{1} & q_{1} & r_{1} \\ p_{2} & q_{2} & r_{2} \\ p_{3} & q_{3} & r_{3} \end{matrix}]

p_{i}, q_{i}, r_{i}

are the cofactors of the elements

l_{i}, m_{i}, n_{i}

for

i = 1,2, 3 .

(l_{1}, m_{1}, n_{1}), (l_{2}, m_{2}, n_{2})

and

(l_{3}, m_{3}, n_{3})

are the direction cosines of three mutually perpendicular lines, then

(p_{1}, q_{1}, r_{1}), (p_{2}, q_{2}, r_{2})

and

(p_{3}, q_{3}, r_{3})

are

MEDIUM

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 3 - Three Dimensional Coordinate System>Three Dimensional Coordinate System Exercise 1>Q 93

A B C D

is a tetrahedron such that each

∆ A B C, ∆ A B D

and

∆ A C D

has a right angle at

A

. If

ar (∆ A B C) = K_{1}

ar (∆ A B D) = K_{2}

and

ar (∆ B C D) = K_{3}

, then

ar (∆ A C D)

HARD

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 3 - Three Dimensional Coordinate System>Three Dimensional Coordinate System Exercise 1>Q 94

In a regular tetrahedron, if the distance between the midpoints of opposite edges is unity, then its volume is

MEDIUM

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 3 - Three Dimensional Coordinate System>Three Dimensional Coordinate System Exercise 1>Q 95

A variable plane makes intercepts on

X, Y

and

Z

-axes and it makes a tetrahedron of volume

64

cu. The locus of foot of perpendicular from origin on this plane is

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JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 3 - Three Dimensional Coordinate System>Three Dimensional Coordinate System Exercise 1>Q 96

P, Q, R, S

are four coplanar points on the sides

A B, B C, C D, D A

of a skew quadrilateral, then

\frac{A P}{P B} \cdot \frac{B Q}{Q C} \cdot \frac{C R}{R D} \cdot \frac{D S}{S A}

equals

MEDIUM

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 3 - Three Dimensional Coordinate System>Three Dimensional Coordinate System Exercise 2>Q 97

Given the equations of the line

3 x - y + z + 1 = 0 & 5 x + y + 3 z = 0

. Then which of the following is correct?

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JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 3 - Three Dimensional Coordinate System>Three Dimensional Coordinate System Exercise 2>Q 98

Consider the family of planes

x + y + z = c

, where

c

is a parameter intersecting the coordinate axes at

P, Q

and

R

and

α, β, γ

are the angles made by each member of this family with positive

x, y & z

axes. Which of the following interpretations hold good for this family?

MEDIUM

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 3 - Three Dimensional Coordinate System>Three Dimensional Coordinate System Exercise 2>Q 99

Equation of the line through the point

(1, 1, 1)

and intersecting the lines

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

and

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

The shortest distance between the two lines L1: x=k1; y=k2 & L2: x=k3; y=k4 is equal to

Important Questions on Three Dimensional Coordinate System

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The shortest distance between the two lines $L_{1} : x = k_{1}; y = k_{2} & L_{2} : x = k_{3}; y = k_{4}$ is equal to