Class 11 NCERT PHYSICS • Chapter 3: Motion in a Plane • Question 5

QUESTION 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.

SOLUTION

We need to evaluate the validity of five statements regarding vector properties, kinematics, and vector addition principles.

Concept: Vector Magnitude

Vector Magnitude: The length of a vector, calculated as $|\vec{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}$ , which is always a scalar.
Vector Components: The projection of a vector onto an axis. While often represented by scalar coefficients, the components of a vector are themselves vectors (e.g., $\vec{A}_x = A_x \hat{i}$ ).
Path Length vs Displacement: Path length is the total distance traveled, whereas displacement is the straight-line vector from the start to the end point.
Average Speed and Velocity: Average speed is $\frac{\text{Path Length}}{\Delta t}$ , and average velocity is $\frac{\text{Displacement}}{\Delta t}$ .
Null Vector: A vector with zero magnitude, denoted as $\vec{0}$ .

Given:

(a) Magnitude of a vector is always a scalar.
(b) Each component of a vector is always a scalar.
(c) Total path length is always equal to the magnitude of displacement.
(d) Average speed $\ge$ magnitude of average velocity.
(e) Three non-coplanar vectors can never add up to a null vector.

Solving:

Statement (a): The magnitude of a vector represents its "size" or "length." By definition, it is a single numerical value (a real non-negative number) and does not have a direction. Therefore, it is always a scalar.
- Result: True.
Statement (b): A vector $\vec{A}$ can be resolved into its components along the coordinate axes. While $A_x, A_y, A_z$ are the scalar magnitudes of these components, the actual components of the vector are $\vec{A}_x = A_x \hat{i}$ , $\vec{A}_y = A_y \hat{j}$ , and $\vec{A}_z = A_z \hat{k}$ , which are vectors. Since they have direction, the statement that they are "always scalars" is incorrect.
- Result: False.
Statement (c): The total path length is the actual distance covered by a particle. The magnitude of displacement is the shortest distance between the initial and final positions. Path length is equal to the magnitude of displacement only if the particle moves in a straight line without reversing direction. In all other cases, path length is greater.
- Result: False.
Statement (d): Let $s$ be the total path length and $|\Delta \vec{r}|$ be the magnitude of displacement. We know that $s \ge |\Delta \vec{r}|$ . Dividing both sides by the time interval $\Delta t$ :

\frac{s}{\Delta t} \ge \frac{|\Delta \vec{r}|}{\Delta t}

Since $\frac{s}{\Delta t}$ is the average speed and $\frac{|\Delta \vec{r}|}{\Delta t}$ is the magnitude of the average velocity, the statement is correct.

Result: True.

Statement (e): For the sum of three vectors $\vec{A} + \vec{B} + \vec{C}$ $A + B + C$ to be a null vector ( $\vec{0}$ $0$ ), the third vector $\vec{C}$ $C$ must be equal to $-(\vec{A} + \vec{B})$ $- (A + B)$ . The resultant of two vectors $\vec{A}$ $A$ and $\vec{B}$ $B$ always lies in the plane containing $\vec{A}$ $A$ and $\vec{B}$ $B$ . Therefore, $\vec{C}$ $C$ must also lie in that same plane. If the three vectors are not coplanar, they cannot satisfy this condition.
- Result: True.

Answer:

(a) True
(b) False
(c) False
(d) True
(e) True