Class 11 NCERT PHYSICS • Chapter 3: Motion in a Plane • Question 4

QUESTION 4

State with reasons, whether the following algebraic operations with scalar and vector physical quantities are meaningful :

(a) adding any two scalars,
(b) adding a scalar to a vector of the same dimensions ,
(c) multiplying any vector by any scalar,
(d) multiplying any two scalars,
(e) adding any two vectors,
(f) adding a component of a vector to the same vector

SOLUTION

We need to evaluate the validity of various algebraic operations involving scalars and vectors based on physical and mathematical principles.

Concept: Principle of Homogeneity

The Principle of Homogeneity states that only physical quantities of the same nature (dimensions and type) can be added or subtracted. Scalars are defined by magnitude only, while vectors possess both magnitude and direction. Multiplication is generally permissible across different types, but addition requires strict compatibility in both dimensionality and mathematical category.

Given:

(a) Addition of two scalars.
(b) Addition of a scalar and a vector of the same dimensions.
(c) Multiplication of a vector by a scalar.
(d) Multiplication of two scalars.
(e) Addition of two vectors.
(f) Addition of a vector component to the same vector.

Solving:

Adding any two scalars
- This is not meaningful in general. While they are both scalars, they must represent the same physical quantity (e.g., you cannot add mass to temperature). Only scalars with the same dimensions can be added.
Adding a scalar to a vector of the same dimensions
- This is not meaningful. A scalar lacks direction, whereas a vector has a specific orientation in space. Even if they share dimensions (e.g., speed and velocity), a scalar cannot be added to a vector because their mathematical natures differ.
Multiplying any vector by any scalar
- This is meaningful. Multiplying a vector $\vec{A}$ by a scalar $k$ results in a new vector $k\vec{A}$ whose magnitude is scaled and whose direction remains the same (or reverses if $k < 0$ ). For example, $\vec{F} = m\vec{a}$ (Newton's Second Law).
Multiplying any two scalars
- This is meaningful. Any two scalars can be multiplied regardless of their nature. The product results in a new scalar quantity with dimensions equal to the product of the original dimensions (e.g., $\text{work} = \text{power} \times \text{time}$ ).
Adding any two vectors
- This is not meaningful in general. Similar to scalars, two vectors can only be added if they represent the same physical quantity (e.g., you cannot add a force vector to a velocity vector). They must have the same dimensions.
Adding a component of a vector to the same vector
- This is meaningful. A component of a vector is itself a vector (e.g., $\vec{A}_x = A_x \hat{i}$ ). Since the component and the original vector represent the same physical quantity and are both vectors, their addition is mathematically and physically valid.

Answer:

(a) Not meaningful; they must represent the same physical quantity.
(b) Not meaningful; a scalar cannot be added to a vector.
(c) Meaningful; it scales the magnitude of the vector.
(d) Meaningful; results in a new scalar quantity.
(e) Not meaningful; they must represent the same physical quantity.
(f) Meaningful; both are vectors of the same physical nature.