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, © 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) No, adding any two scalars is not meaningful due to only the scalars of same dimensions, (i.e., of same nature) can be added.
(b) No, adding a scalar to a vector of the same dimension is not meaningful due to a scalar cannot be added to a vector.
© Yes, multiplying any vector by any scalar is
meaningful in algebraic operation. It is due to when any vector is multiplied by any scalar, then we get a vector having magnitude equal to scalar number of times the magnitude of the given vector. For example, when acceleration a is multiplied by mass m, we get force F = ma which is a meaningful operation. .
(d) Yes, it is clear product of two scalar gives a meaningful result. For example, when power P is multiply by time t, then we get work done (W), i.e., W = Pt, which is a useful algebraic operation.
(e) No, as the two vectors of same dimensions (i.e., of the same nature) can only be added, so addition of any two vectors is not a meaningful algebraic operation.
(f) Addition of a component of a vector to the same vector can be done by the law of vector addition. So algebraic operation is not a meaningful operation