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,
$(b)$ 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.
$(b)$ 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.
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