Question

State with reasons, whether the following algebraic operations with scalar and 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.

Answer 1

(A) Addition of scalars.
The addition of two scalar quantities is meaningful only if they both represent the same physical quantity.
Like mass and volume cannot be added.
(B) Adding a scalar to a vector of the same dimensions.
Same type of quantities can be added.
The addition of a vector quantity with a scalar quantity is not meaningful. A scalar cannot be added to a vector.
Like speed and velocity cannot be added.
(C) Multiplying any vector by any scalar.
Recall multiplication of scalar and vector quantities.
Any vector can be multiplied by a scalar.
A vector when multiplied by a scalar quantity can give us a vector quantity.
When the vector quantity acceleration is multiplied with mass (scalar), we get force,
$\overrightarrow{F}=m\overrightarrow{a}$ which is meaningful.
(D) Multiplying any two scalars.
Recall multiplication of scalar quantities.
A scalar which represents a physical quantity can be multiplied with another scalar having same or different dimension.
For example, when power is multiplied with time, we get work done.
(E) Adding any two vectors.
The vector quantities can only be added by vector law of addition.
The sum of two or more vector called the resultant. The resultant of two vector can be found using either the parallelogram method or triangle method. So algebraic sum of two vectors is meaningless.
(F) Adding a component of a vector to the same vector.
The vector quantities can only be added by vector law of addition.
A component of a vector can be added to the same vector only by using law of vector addition. So, the addition of a component of a vector to the same vector is not a meaningful algebraic operation.