MEDIUM

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Consider three planes P₁, P₂, P₃.

P₁ : ax + by + cz = 0

P₂ : bx + cy + az = 0

P₃ : cx + ay + bz = 0

Now give the answers of the following questions.

If planes meet in a common line and $\vec{A} = a \hat{j} + b \hat{j} + c \hat{k}, \vec{B} = b \hat{i} + c \hat{j} + a \hat{k}, \vec{C} = c \hat{i} + a \hat{j} + b \hat{k}$ , then the value of $|\begin{matrix} \vec{A} . \vec{A} & \vec{A} . \vec{B} & \vec{A} . \vec{C} \\ \vec{B} . \vec{A} & \vec{B} . \vec{B} & \vec{B} . \vec{C} \\ \vec{C} . \vec{A} & \vec{C} . \vec{B} & \vec{C} . \vec{C} \end{matrix}|$ is equal to

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Important Questions on Vectors

HARD

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

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:

HARD

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

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 :

HARD

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

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}

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

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>Coordinate Geometry>Vectors>Scalar TripIe Product

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

EASY

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

Which of the following is not equal to

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

HARD

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

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:

HARD

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

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 :

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

\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>Coordinate Geometry>Vectors>Scalar TripIe Product

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

EASY

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

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.

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

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}

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

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

HARD

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

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.

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

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>Coordinate Geometry>Vectors>Scalar TripIe Product

h

is the altitude of a parallelopiped determined by the vectors

\hat{a}, \hat{b}, \hat{c}

and the base is taken to be the parallelogram determined by

\hat{a}

and

\hat{b}

where

\hat{a} = \hat{i} + \hat{j} + \hat{k}, \hat{b} = 2 \hat{i} + 4 \hat{j} - \hat{k}

and

\hat{c} = \hat{i} + \hat{j} + 3 \hat{k},

then the value of

19 h^{2}

HARD

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

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>Coordinate Geometry>Vectors>Scalar TripIe Product

[\vec{a} \times \vec{b} \vec{b} \times \vec{c} \vec{c} \times \vec{a}] = λ {[\vec{a} \vec{b} \vec{c}]}^{2}

then

λ

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

If the origin and the points

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

and

R (x, y, z)

are co-planar then

MEDIUM

Mathematics>Coordinate Geometry>Vectors>Scalar TripIe Product

Let $\vec{a}, \vec{b} and \vec{c}$ be three non - zero vectors such that no two of them are collinear and $(\vec{a} \times \vec{b}) \times \vec{c} = \frac{1}{3} |\vec{b}| |\vec{c}| \vec{a}$ . If $θ$ is the angle between vectors $\vec{b} and \vec{c}$ , then a value of $\sin θ$ is

Important Questions on Vectors

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