Difference between Scalar and Vector

Scalar quantity

The physical quantity that has only magnitude and no direction is scalar. A scalar quantity is a one-dimensional quantity because it does not have any direction to make it two or three-dimensional. Scalar is just a number with some value to measure objects or quantities. The resultant of two or more scalar will always be a scalar quantity. Algebraic operations are possible only when two scalar quantities have the same units.
Examples of scalar quantity are – speed, time, mass, electric charge, volume, temperature, gravitational force, etc.

Vector quantity

The physical quantity that has both magnitude and direction is called a vector quantity. Because vector quantity has direction, it can be two or three-dimensional. Vector algebra is the method to solve vector quantities. It is represented by an arrow over the top of the vector number. The resultant of two vectors can be a scalar or a vector quantity depending upon the algebraic function we have applied.

A dot product of two vectors will result in a scalar quantity, whereas a cross product will result in a vector quantity. Vectors can be resolved into sine and cosine functions to find the direction of the resultant vector. The sine function is used in the dot product of vector quantities, and the cosine function is used in the cross product of the vector quantities.
Examples of vector quantity are – displacement, velocity, electric field, acceleration, polarization, force, linear momentum, etc.

Similarities between scalar and vector

Both scalar quantities and vector quantities have a certain finite magnitude.
Both scalar quantity and vector quantity express specific physical quantities.
Both scalar quantity and vector quantity express specific physical quantities.
Both scalar quantities and vector quantities are measurable (can be measured using a suitable instrument) and quantifiable (can be expressed as numerical magnitude with the corresponding unit of measurement).

Difference between scalar and vector quantity

Scalar quantity	Vector quantity
Scalar quantity has only magnitude.	Vector quantity has only direction.
It is always one dimensional.	It can be two or three dimensional.
The quantity changes if the magnitude changes.	The quantity changes only there is a change in both magnitude and direction.
Algebraic rules can be followed on scalar quantities.	Special rules, like dot and cross products, are followed by vector quantities.
One scalar quantity can divide, multiply with another scalar, and the result will be a scalar only.	One vector cannot operate simply like a scalar over another vector.
Scalar quantity does not deal with any angles because it does not have any direction.	Vector quantities deal with angles because they are two or dimensional. Dot product uses sine angle to find the direction of the resultant. Whereas, cross product uses cosine angle to find the direction of the resultant vector.
Examples of scalar quantity are – speed, time, mass, electric charge, volume, temperature, gravitational force, etc.	Examples of vector quantity are – displacement, velocity, electric field, acceleration, polarization, force, linear momentum, etc.
The resultant of two scalar quantities will be a scalar quantity only.	The resultant of two vector quantities can either be a scalar quantity or a vector quantity depending upon the type of algebraic operation.

Example

To reach his home, a boy walks 10 m north, 12 m east, 3 m west, and 5 m south. What is the magnitude of his displacement from his original point?

Answer: We know displacement is a vector quantity because it has both magnitude and direction. The boy also walks in both positive and negative directions if we consider his starting point as origin.
To find the total distance traveled along the y-axis, let us consider the movement towards the north to be positive and the south negative.
∑y = 10m − 5m = 5m
He moved a net of 5 meters to the north along the y-axis.
In the same way, we can consider the boy’s movement towards the east to be positive and the west to be negative.
∑y = −3m + 12m = 9m
He moved a net of 9 m to the east.
Now using Pythagoras theorem, we can find the resultant as-
D2 = (∑x2) + (∑y2)
Substituting the values, we get,
D2 = 92 + 52
This gives, D = 10.30m.

Frequently asked questions on scalar and vector quantity

Ques: Is force a scalar or vector quantity?
Ans: Force has both magnitude and direction. Hence, it is a vector quantity. Other forces, such as gravitational force, are a scalar quantity as they only have magnitude and no direction.

Ques: Is time a vector quantity of scalar quantity?
Ans: Time is a scalar quantity. It has only magnitude and no direction. Therefore, it never changes its course and moves only in one direction (not a vector direction).