#### Question

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 1

a) Meaningful

(b) Not Meaningful

(c) Meaningful

(d) Meaningful

(e) Meaningful

(f) Meaningful

**Explanation:**

**(a)**The addition of two scalar quantities is meaningful only if they both represent the same physical quantity.

**(b)**The addition of a vector quantity with a scalar quantity is not meaningful.

**(c) **A scalar can be multiplied with a vector. For example, force is multiplied with time to give impulse.

**(d) **A scalar, irrespective of the physical quantity it represents, can be multiplied with another scalar having the same or different dimensions.

**(e) **The addition of two vector quantities is meaningful only if they both represent the same physical quantity.

**(f) **A component of a vector can be added to the same vector as they both have the same dimensions.

#### Solution 2

(a) No, because only the scalars of same dimensions can be added.]

(b) No, because a scalar cannot be added to a vector.

(c) Yes, multiplying a vector with a scalar gives the scalar (number) times the vector quantity which makes sense and one gets a bigger vector. For example, when acceleration A is multiplied by mass m, we get a force F = ml

(d) Yes, two scalars multiplied yield a meaningful result, for example multiplication of rise in temperature of water and its mass gives the amount of heat absorbed by that mass of water.

(e) No, because the two vectors of same dimensions can be added.

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