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.

(N/A) Meaningful: The addition of two scalar quantities is meaningful only if they both represent the same physical quantity.
$(b)$ Not Meaningful: The addition of a vector quantity with a scalar quantity is not meaningful because they represent different types of physical entities.
$(c)$ Meaningful: $A$ scalar can be multiplied with a vector. For example,force is multiplied with time to give impulse $(I = F \cdot \Delta t)$.
$(d)$ Meaningful: $A$ scalar,irrespective of the physical quantity it represents,can be multiplied with another scalar having the same or different dimensions.
$(e)$ Meaningful: The addition of two vector quantities is meaningful only if they both represent the same physical quantity.
$(f)$ Meaningful: $A$ component of a vector can be added to the same vector as they both have the same dimensions and represent the same physical quantity.