State with reasons, whether the following algebraic operations wi… - Vidyadip

Question

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.

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Answer

(a) No, because two scalars of same nature can be added.
(b) No, adding vector of same dimension in a scalar is not meaningful because scalars of same dimensions can be added. A scalar and a vector can't be added.
(c) Multiplication of a vector by a scalar is possible. It is an algebraic process because when we multiply a vector with a scalar we get a scalar times vector, which gives the magnitude of vector. That is, if we mutliply acceleration $(\vec{a})$ by mass $(m)$, we get force $\overrightarrow{ F }=m \vec{a}$, which is a meaningful process.
(d) Yes, multiplication of two scalars gives a meaningful result. Since, we multiply power P by time $(t)$, we get work $(w)= P \times t$ which is meaningful algebraic process.
(e) No, because both vectors can be added if they are of same nature.
(f) Yes, in a vector, its components can be added becuase both the vectors are of same nature, i.e. having the same dimension.

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