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Question-1

What is the difference between (+) and (-)?

Solution:
Magnitude is same, but direction is opposite.

Question-2

Define equal vectors.

Solution:
Two vectors are said to be equal vectors if they have equal magnitudes and same direction.

Question-3

What do you mean by null vector?

Solution:
A vector whose magnitude is zero and has no sense of direction is called null vector.

Question-4

When is the sum of two vectors maximum and when is it minimum?

Solution:
When both are in the same direction, the sum of the vectors is a maximum and when in opposite directions the sum of the two vectors is minimum.

Question-5

Under what conditions will the directions of sum and difference of two vectors be same?

Solution:
The directions of sum and difference of two vectors will be the same when they are unequal in magnitude and are in the same direction.

Question-6

(f) Adding a component of a vector to the same vector.

 


Solution:
(a) Addition of any two scalars is not a meaningful algebraic operation because they can be added only when their nature is same, e.g., speed cannot be added to velocity.

(b) Addition of a scalar to a vector is not allowed even though they have the same dimension because a vector is a directed quantity while a scalar has no direction e.g., speed cannot be added to velocity.

(c) Multiplication of any vector by a scalar is a meaningful algebraic operation =mv2 i.e., mass (scalar) multiplied by velocity (vector) gives rise to momentum.

(d) Yes, when power P is multiplied by time t, we get

work done= Pt, which is a useful operation.

(e) No, because the two vectors of same dimensions cannot be added.

(f) Yes, because both are vectors of the same dimension.

Question-7

Does the nature of a vector change when it is multiplied by a scalar?

Solution:
The nature of a vector may or may not be changed when it is multiplied by a scalar. For example, when a vector is multiplied by a pure number (like 1,2,3,4,….etc) then the nature of the vector does not change. On the other hand, when a vector is multiplied by a scalar physical quantity, then the nature of the vector changes. For example, when an acceleration (vector) is multiplied by a mass (scalar) of a body, it gives a force (vector quantity) whose nature is different from acceleration.

Question-8

Does a vector have a location in space in addition to the magnitude and direction? Can two equal vectors and at different locations in space necessarily have an identical physical effect?

Solution:
Yes, each vector has a location in space in addition to magnitude and direction. Two equal vectors and having different locations may not have the same physical effect.

Question-9

A vector has both magnitude and direction. Does that mean-anything that has magnitude and direction is necessarily a vector? The rotation of a body can be specified by the direction of the axis of rotation, and the angle of rotation about the axis. Does that make any rotation of a vector?

Solution:
Generally rotation is not considered a vector, though it has magnitude and direction. The reason is that addition of two finite rotations does not obey commutative law.  Since, addition of vectors should obey commutative law, a finite rotation cannot be regarded as a vector. However, infinitely small rotations obey commutative law for addition and hence an infinitely small rotation is a vector.

Question-10

Do + and - lie in the same plane? Explain.

Solution:
Yes, + is represented by the diagonal of the parallelogram drawn with and as adjacent sides. The diagonal passes through the common tail of and . However, - is represented by the diagonal of the same parallelogram not passing through the common tail of and.

Question-11

Can three vectors lying in a plane add up to give a null vector? If yes, state the necessary conditions?

Solution:
Yes, it is possible. The necessary conditions is that the resultant of two vectors must have a magnitude equal to the magnitude of the third vector but opposite in direction.

Question-12

What is the angle between two vectors if the ratio of their dot product and cross product is 3?

Solution:

or tan = or θ = 30°

Question-13

When the component of a vector A along the direction of B is zero, what can you conclude about the two vectors ?

Solution:
We have to consider two vectors A and B such that the angle between them is θ.

Question-14

What are the conditions for two vectors to be (i) parallel and (ii) perpendicular to each other?

Solution:
(i) We know that, A × B =

If two vectors are parallel, i.e. θ = 0, then = 0

i.e., if two vectors are parallel, the product must be zero.

(ii) Also = A B cos θ

If two vectors are perpendicular, i.e. θ = 90° ,

then = 0 i.e. if two vectors are perpendicular their dot product must be zero.

Question-15

What is the effect on the magnitude of the resultant of two vectors when the angle between the two vectors is increased from 0° to π?

Solution:
If and be the two given vectors, then the magnitude of their resultant is given by

R =
when
θ is increased from 0 to π , cos θ goes on decreasing. Therefore, the magnitude of the resultant will also go on decreasing.

Question-16

Which of the following quantities are independent of the choice of orientation of the coordinate axes , 3Ax + 2By, ||, angle between and and λ , where λ is a scalar quantity.

Solution:
In a vector, its magnitude and angle two vectors do not depend upon the choice of orientation of the coordinate axes. Therefore, , 3Ax + 2By, ||, angle between and and λ are independent or orientation of 'coordinate axes. Because the quantity 3Ax + 2By depends upon the magnitude of components x and y-axis, it will change with change of coordinate axes.

Question-17

Can the components of a vector have magnitude greater than the vector itself? Can a rectangular component do so? Explain.

Solution:
Yes, component of a vector cannot have magnitude greater than that of vector itself. The same is true in the case of rectangular component also. The rectangular components of a vector are: Ax = A cos θ and Ay = A sin θ . As maximum values of both sin θ and cos θ are equal to one, hence ||or || cannot be greater than ||.




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