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) If the scalars are similar type of quantities, adding of scalars will be meaningful. Scalars of the same dimension can only be added.
(b) Adding a scalar to a vector of the same dimension is meaningless. a scalar cannot be added to a vector.
(c) Any vector can be multiplied by a scalar. A vector when multiplied by a scalar quantity will give us a vector quantity. When the vector quantity acceleration is multiplied by m, we get a force, , which is a meaningful operation.
(d) Two scalars can be multiplied together. For example, when power is multiplied by time t, we get the quatity, work done.
(e) Adding any two vectors is meanigless because two vectors of the same dimnsion can be added.
(f) Adding a component of a vector to the same vector is meaningful. Because both the vectors are of the same dimensions.