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.

Answer 1

(a)

Addition of scalar quantities is possible only if both quantities have the same dimensions

(b)

Adding a scalar to a vector of the same dimension is not possible because a scalar quantity has only magnitude but avector quantity has both direction and magnitude.

(c)

Multiplying any vector by a scalar is possible because when a vector quantity is multiplied by a scalar quantity, the resultant is also a vector.

Consider a velocity vector V → and a scalar quantity mass ( m ). The multiplication of these two quantities gives linear momentum p → which is a vector quantity.

(d)

Multiplication of any two scalars is possible and it gives another scalar quantity.

Consider a liquid of mass m and increase in temperature ΔT.Both these quantities are scalar. Multiplication of these quantities gives the amount of heat absorbed by the liquid which is a scalar quantity.

(e)

Addition of vector quantities is possible only if both the quantities have the same dimensions.

(f)

Adding a component of a vector to the same vector is possible because both vectors have the same dimensions.