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.

(a) Addition of two scalars is meaningful since both follow general algebraic laws.

(b) Addition of a scalar to a vector of the same dimensions is not meaningful since the scalar follows general algebraic laws but the vector does not.

(c) Multiplying any vector by a scalar is meaningful as the magnitude of the vector is multiplied by the scalar. For example, force which is a vector when multiplied by the scale time gives the vector impulse.

(d) Multiplying any two scalars is meaningful irrespective of their dimensions. For example, speed multiplied by time gives the distance travelled. Here, speed and time are scalars which give distance which is also a scalar.

(e) Addition of any two vectors is meaningful only if they have the same dimensions. This addition takes places according to vector algebra.

(f) Adding a component of a vector to the same vector is meaningful because both of them have the same dimensions.

NOTE: A scalar quantity is a quantity that is described by a magnitude or numerical value only. A vector quantity is a quantity that is described by both magnitude and direction.