If ( ( a → × b → ) × ( c → × d → ) ) · ( a → × d → ) = 0 , then which of the following is always true?

Either b → or c → must lie in the plane of a → and d →

a → , b → , c → , d → are necessarily coplanar

Either a → or d → must lie in the plane of b → and c →

Either a → or b → must lie in the plane of c → and d →

HARD

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The length of two opposite edges of a tetrahedron are $12$ and $15$ units and the shortest distance between them is $10$ units. If the volume of the tetrahedron is $200$ cubic units, then the angle between the $2$ edges is

50% studentsanswered this correctly

Important Questions on Vector and Matrix Quantities

HARD

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

((\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d})) \cdot (\vec{a} \times \vec{d}) = 0

, then which of the following is always true?

HARD

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

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

are mutually perpendicular vectors having magnitude

1, 2, 3

respectively, then

[\vec{a} + \vec{b} + \vec{c} \vec{b} - \vec{a} \vec{c}]

HARD

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

Let

\vec{a}, \vec{b}

and

\vec{c}

be three unit vectors such that

\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\sqrt{3}}{2} (\vec{b} + \vec{c}) .

\vec{b}

is not parallel to

\vec{c}

, then the angle between

\vec{a}

and

\vec{b}

EASY

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

Which of the following is not equal to

\vec{w} . (\vec{u} \times \vec{v})

MEDIUM

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

Let

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

be three vectors having magnitudes

1, 1

and

2

respectively. If

\vec{a} \times (\vec{a} \times \vec{c}) + \vec{b} = \vec{0},

then the angle between

\vec{a}

and

\vec{c}

HARD

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

A

B

C

and

D

are

(3, 7, 4), (5, - 2, - 3), (- 4, 5, 6)

and

(1, 2, 3)

respectively, then the volume of the parallelepiped with

A B

A C

and

A D

as the coterminous edges, is ….cubic units.

HARD

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

Let

\vec{u}

be a vector coplanar with the vectors

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

and

\vec{b} = \hat{j} + \hat{k}

. If

\vec{u}

is perpendicular to

\vec{a}

and

\vec{u} \cdot \vec{b} = 24

, then

{|\vec{u}|}^{2}

is equal to:

MEDIUM

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

Let

x_{0}

be the point of local maxima of

f (x) = \vec{a} \cdot (\vec{b} \times \vec{c}),

where

\vec{a} = x \hat{i} - 2 \hat{j} + 3 \hat{k}, \vec{b} = - 2 \hat{i} + x \hat{j} - \hat{k}

and

\vec{c} = 7 \hat{i} - 2 \hat{j} + x \hat{k} .

Then the value of

\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}

x = x_{0}

is:

MEDIUM

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

If the origin and the points

P (2, 3, 4), Q (1, 2, 3)

and

R (x, y, z)

are co-planar then

HARD

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

Let

\vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{c} = \hat{j} - \hat{k}

and a vector

\vec{b}

be such that

\vec{a} \times \vec{b} = \vec{c}

and

\vec{a} \cdot \vec{b} = 3

. Then

|\vec{b}|

equals

MEDIUM

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

Let the volume of a parallelepiped whose coterminous edges are given by

\vec{u} = \hat{i} + \hat{j} + λ \hat{k}, \vec{v} = \hat{i} + \hat{j} + 3 \hat{k}

and

\vec{w} = 2 \hat{i} + \hat{j} + \hat{k}

, be

1

cu. unit. If

θ

be the angle between the edges

\vec{u}

and

\vec{w},

then the value of

\cos θ

can be

HARD

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

Let

\vec{a}, \vec{b}

and

\vec{c}

be three unit vectors, out of which vectors

\vec{b}

and

\vec{c}

are non-parallel. If

α

and

β

are the angles which vector

\vec{a}

makes with vectors

\vec{b}

and

\vec{c}

respectively and

\vec{a} \times (\vec{b} \times \vec{c}) = \frac{1}{2} \vec{b}

, then

|α - β|

is equal to :

EASY

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

Let

\vec{a} = \hat{i} - 2 \hat{j} + \hat{k}

and

\vec{b} = \hat{i} - \hat{j} + \hat{k}

, be two vectors. If

\vec{c}

is a vector such that

\vec{b} \times \vec{c} = \vec{b} \times \vec{a}

and

\vec{c} \cdot \vec{a} = 0,

then

\vec{c} \cdot \vec{b}

is equal to.

HARD

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

If the volume of parallelepiped formed by the vectors

\hat{i} + λ \hat{j} + \hat{k}, \hat{j} + λ \hat{k}

and

λ \hat{i} + \hat{k}

is minimum, then

λ

is equal to:

MEDIUM

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

[\vec{a} + 2 \vec{b} - \vec{c} \vec{a} - \vec{b} \vec{a} - \vec{b} - \vec{c}]

HARD

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

If the volume of a parallelepiped whose coterminous edges are

\vec{a} = \hat{i} + \hat{j} + 2 \hat{k},

\vec{b} = 2 \hat{i} + λ \hat{j} + \hat{k}

and

\vec{c} = 2 \hat{i} + 2 \hat{j} + λ \hat{k}

35

cu . m,

then a value of

\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} - \vec{c} \cdot \vec{a}

HARD

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

If the volume of a parallelopiped, whose coterminous edges are given by the vectors

\vec{a} = \hat{i} + \hat{j} + n \hat{k}

\vec{b} = 2 \hat{i} + 4 \hat{j} - n \hat{k}

and,

\vec{c} = \hat{i} + n \hat{j} + 3 \hat{k}

(n \geq 0)

158

cubic units, then :

EASY

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

If the vectors

4 \hat{i} + 11 \hat{j} + m \hat{k}, 7 \hat{i} + 2 \hat{j} + 6 \hat{k}

and

\hat{i} + 5 \hat{j} + 4 \hat{k}

are coplanar, then

m

is equal to

MEDIUM

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

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

and

\vec{c} = \hat{i} + m \hat{j} + n \hat{k}

are three coplanar vectors and

|\vec{c}| = \sqrt{6},

then which one of the following is correct?

EASY

Mathematics>Number and Quantity>Vector and Matrix Quantities>Multiply a vector by a scalar

\vec{a} = (\hat{i} + \hat{j} + \hat{k}),

\vec{a} \cdot \vec{b} = 1

and

\vec{a} \times \vec{b} = \hat{j} - \hat{k}

, then

\vec{b}

The length of two opposite edges of a tetrahedron are 12 and 15 units and the shortest distance between them is 10 units. If the volume of the tetrahedron is 200 cubic units, then the angle between the 2 edges is

Important Questions on Vector and Matrix Quantities

Learn and Prepare for any exam you want !

The length of two opposite edges of a tetrahedron are $12$ and $15$ units and the shortest distance between them is $10$ units. If the volume of the tetrahedron is $200$ cubic units, then the angle between the $2$ edges is