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) Adding any two scalars
Answer: Meaningful only if they represent the same physical quantity.
Scalars with the same dimensions (e.g. 3 kg + 5 kg) can be added to give a physically meaningful result. Adding scalars of different kinds (e.g. 3 kg + 5 s) is meaningless because they represent different physical quantities.
(b) Adding a scalar to a vector of the same dimensions
Answer: Not meaningful.
A scalar has only magnitude, while a vector has both magnitude and direction, so they are different types of quantities. Even if their units match, addition of a scalar and a vector is undefined in vector algebra.
(c) Multiplying any vector by any scalar
Answer: Meaningful.
Multiplying a vector by a scalar scales the magnitude of the vector by that factor while keeping (or reversing, if scalar is negative) its direction. This is a standard vector operation, e.g. \( \vec{F} = m \vec{a} \) or \( \vec{p} = m \vec{v} \).
(d) Multiplying any two scalars
Answer: Meaningful.
The product of two scalars is another scalar; dimensions may change but the operation is well-defined. For example, speed (m/s) × time (s) = distance (m); mass × specific heat = heat capacity, etc.
(e) Adding any two vectors
Answer: Meaningful only if they represent the same physical quantity.
Vector addition is defined for vectors of the same kind (e.g. two forces, two displacements). Adding vectors of different physical quantities (e.g. a force vector and a velocity vector) is physically meaningless, even though algebraically one can form a formal sum.
(f) Adding a component of a vector to the same vector
Answer: Meaningful.
A component of a vector is itself a vector of the same kind (e.g. x-component of force is a force). Adding this component to the original vector is just vector addition of two like vectors, giving a new vector (same direction as the original if the component is along it).