# NCERT Class 11 Physics Solutions: Chapter 4 – Motion in a Plane Part 2

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Question 4.5:

Read each statement below carefully and state with reasons, if it is true or false:

(a) The magnitude of a vector is always a scalar.

(b) Each component of a vector is always a scalar.

(c) The total path length is always equal to the magnitude of the displacement vector of a particle.

(d) The average speed of a particle (defined as total path length divided by the time taken to cover the path) is either greater or equal to the magnitude of average velocity of the particle over the same interval of time.

(e) Three vectors not lying in a plane can never add up to give a null vector.

(a) True: Because the magnitude of a vector is a number. Hence, it is a scalar.

(b) False: Because each component of a vector is also a vector.

(c) False: Because Total path length is a scalar quantity, whereas displacement is a vector quantity. Hence, the total path length is always greater than the magnitude of displacement. It becomes equal to the magnitude of displacement only when a particle is moving in a straight line.

(d) True: It is because of the fact that the total path length is always greater than or equal to the magnitude of displacement of a particle.

(e) True: three vectors, which do not lie in a plane, cannot be represented by the sides of a triangle taken in the same order.

Question 4.6:

Establish the following vector inequalities geometrically or otherwise:

(a)

(b)

(c)

(d)

When does the equality sign above apply?

(a) Let two vectors and be represented by the adjacent sides of a parallelogram OMNP, as shown in the given figure.

Here, we can write:

In a triangle, each side is smaller than the sum of the other two sides. Therefore, inwe have:

ON < (OM+MN)

If the two vectors and act along a straight line in the same direction, then we can write:

Combining equations (iv) and (v), we get:

(b) Let two vectors and be represented by the adjacent sides of a parallelogram OMNP, as shown in the given figure.

Here, we have:

In a triangle, each side is smaller than the sum of the other two sides. Therefore, in OMN, we have:

If the two vectors and act along a straight line in the same direction, then we can write:

Combining equations (iv) and (v), we get:

(c) Let two vectors and be represented by the adjacent sides of a parallelogram PORS, as shown in the given figure.

Here we have:

In a triangle, each side is smaller than the sum of the other two sides. Therefore, in ΔOPS, we have:

If the two vectors act in a straight line but in opposite directions, then we can write:

Combining equations (iii) and (iv), we get:

(d) Let two vectors and be represented by the adjacent sides of a parallelogram PORS, as shown in the given figure.

The following relations can be written for the given parallelogram.

…( ii )

The quantity on the LHS is always positive and that on the RHS can be positive or negative. To make both quantities positive, we take modulus on both sides as:

If the two vectors act in a straight line but in the opposite directions, then we can write:

Combining equations (iv) and (v), we get: