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QUESTION: 1

Vector has

Solution:

A vector has both magnitude as well as direction.

QUESTION: 2

Correct form of distributive law is

Solution:

Distributive law is given by :

QUESTION: 3

Magnitude of the vector

Solution:

We have :

QUESTION: 4

Find the unit vector in the direction of vector where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively

Solution:

QUESTION: 5

If is a non zero vector of magnitude ‘a’ and λ a non zero scalar, then λ is a unit vector if

Solution:

λ is a unit vector if and only if is equal to

QUESTION: 6

Find the values of x and y so that the vectors are equal

Solution:

QUESTION: 7

If P_{1}(x_{1}, y_{1}, z_{1}) and P_{2}(x_{2}, y_{2}, z_{2}) are any two points, then the vector joining P1 and P2is the vector P1P2. Magnitude of the vector

Solution:

If P_{1}(x_{1}, y_{1}, z_{1}) and P_{2}(x_{2}, y_{2}, z_{2}) are any two points, then the vector joining P1 and P2is the vector P1P2, then ;

QUESTION: 8

Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (– 5, 7).

Solution:

The scalar and vector components of the vector with initial point (2, 1) and terminal point (– 5, 7) is given by : (- 5 – 2) i.e. – 7 and (7 – 1) i.e. 6. Therefore, the scalar components are – 7 and 6 .,and vector components are

QUESTION: 9

Find a vector in the direction of the vector which has a magnitude of 8 units

Solution:

QUESTION: 10

Find , if and

Solution:

QUESTION: 11

Direction angles are angles

Solution:

α,β,γ are the angles which the position vector makes with the positive x-axis ,y-axis and z-axis respectively are called direction angles.

QUESTION: 12

are any three vectors then the correct expression for distributivity of scalar product over addition is

Solution:

are any three vectors then the correct expression for distributivity of scalar product over addition is :

QUESTION: 13

Find the values of x and y so that the vectors

Solution:

QUESTION: 14

Find the direction cosines of the vector

Solution:

QUESTION: 15

Find a unit vector perpendicular to each of

Solution:

It is given that:

Therefore, the unit vector perpendicular to both the vectors and

QUESTION: 16

Direction cosines

Solution:

Cosines of the angles α,β,γ are called direction cosines.

QUESTION: 17

Magnitude of the vector

Solution:

We have :

QUESTION: 18

Find the sum of the vectors and

Solution:

We have:

QUESTION: 19

Find the direction cosines of the vector joining the points A(1, 2, –3) and B(–1, –2, 1), directed from A to B.

Solution:

Therefore, the D.C.’s of vector AB are given by:

QUESTION: 20

If a unit vector makes angles π/3 with and an acute angle θ with then find θ

Solution:

QUESTION: 21

If l, m and n are direction cosines of the position vector OP the coordinates of P are

Solution:

If l , m and n are the direction cosines of vector then , the coordinates of point P are given by : lr ,mr and nr respectively.

QUESTION: 22

Unit vectors along the axes OX, OY and OZ are denoted by

Solution:

represents the unit vectors along the co ordinate axis i.e. OX ,OY and OZ respectively.

QUESTION: 23

Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

Solution:

QUESTION: 24

Find the angle between two vectors with magnitudes and 2, respectively, having

Solution:

QUESTION: 25

If a unit vector makes angles and an acute angle θ with , then the components of are

Solution:

Let It is given that left| , then ,

Putting these values in (1) , we get :

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