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.

$(a)$ 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.

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

$(d)$ 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.