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

Magnitude is same, but direction is opposite.


Define equal vectors.

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


What do you mean by null vector?

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


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

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.


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

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.


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


(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.


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

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.


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?

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.


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?

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.


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

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.


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

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.


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


or tan = or θ = 30°


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

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


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

(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.


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?

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

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


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.

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.


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

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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