Scalar and Vector

In physics, quantities like force, speed, velocity, and work are categorised as either scalar or vector quantities. A scalar quantity is one that only has magnitude, such as mass or temperature. Conversely, a vector quantity is characterised by both magnitude and direction, as seen in quantities like force or velocity. This distinction plays a key role in describing motion and interactions in physical systems. Here, we will explore scalar and vector quantities, their properties, and their role in describing physical phenomena like motion, forces, and energy, helping us analyse real-world situations accurately.

Scalar Quantities Definition

A scalar quantity is a physical quantity that has only magnitude and no direction.

• It is described by a single numerical value, typically with units.
• Scalars represent quantities where direction is not relevant, such as temperature, mass, time, and energy.
• They can be added, subtracted, multiplied, and divided using standard arithmetic operations.
• Scalars are easier to work with compared to vector quantities since they don’t require considering directions or components.

Examples of Scalar Quantities

There are below some examples of scalar quantities:

• Temperature
• Mass
• Time
• Distance
• Speed
• Energy
• Area
• Volume
• Density
• Electric Charge

These quantities can be measured using instruments such as thermometers, scales, stopwatches, rulers, speedometers, and wattmeters.

For example, if a car travels 100 kilometers in 2 hours, its average speed can be calculated as 50 kilometers per hour (km/h) by dividing the distance traveled by the time taken.

Vector Quantities Definition

A vector quantity is a physical quantity that has both magnitude and direction.

• It is described by both a numerical value and a specified direction, often represented as an arrow or in component form.
• Vectors are used to represent quantities where direction is important, such as displacement, velocity, force, and acceleration.
• They can be added, subtracted, and multiplied using vector-specific operations, such as the dot product and cross product.
• Vectors require consideration of both magnitude and direction, making them more complex to work with compared to scalar quantities.

Examples of Vector Quantities

There are below some examples of vector quantities:

• Velocity
• Acceleration
• Force
• Displacement
• Momentum
• Thrust
• Pressure
• Electric Field
• Magnetic Field
• Weight
• Impulse
• Drag Force
• Lift Force
• Angular Velocity
• Linear Momentum

Vector quantities are used in many fields of science and engineering, such as mechanics, electromagnetism, fluid dynamics, and quantum mechanics. They are essential for describing the behavior of physical systems and making predictions about their future states.

For example, if a car is traveling at a velocity of 50 km/h towards the east, its velocity can be represented as a vector with an arrow pointing to the right (east) and a length of 50 km/h.

Also Read, Difference between Scalar and Vectors

Vector Notation

Vector notation is a way or notation used to represent a quantity that is a vector, through an arrow (⇢) above its symbol, as shown below:

Equality of Vectors

Two vectors are considered equal when they have the same magnitude and the same direction. The figure below shows two vectors that are equal; notice that these vectors are parallel to each other and have the same length. The second part of the figure shows two unequal vectors, which, even though they have the same magnitude, are not equal because they have different directions.

Multiplication of Vectors with Scalar

Multiplying a vector a with a constant scalar k gives a vector whose direction is the same but the magnitude is changed by a factor of k. The figure shows the vector after and before it is multiplied by the constant k. In mathematical terms, this can be rewritten as, 

|k\vec{v}| = k|\vec{v}| 

if k > 1, the magnitude of the vector increase while it decreases when the k < 1. 

Addition of Vectors

Vectors cannot be added by usual algebraic rules. While adding two vectors, the magnitude and the direction of the vectors must be taken into account.

Triangle law is used to add two vectors, the diagram below shows two vectors "a" and "b" and the resultant is calculated after their addition. Vector addition follows commutative property, this means that the resultant vector is independent of the order in which the two vectors are added. 

\vec{a} + \vec{b} = \vec{c}
\vec{a} + \vec{b} = \vec{b} + \vec{a}      - (Commutative Property)

Triangle Law of Vector Addition

Consider the vectors given in the figure above. The line PQ represents the vector "p", and QR represents the vector "q". The line QR represents the resultant vector. The direction of AC is from A to C.  

Line AC represents, 

\vec{p} + \vec{q}

The magnitude of the resultant vector is given by, 

\sqrt{|p|^2 + |q|^2 + 2|p||q|cos(\theta)}

θ represents the angle between the two vectors. Let φ be the angle made by the resultant vector with the vector p.

tan (\phi) = \dfrac{q\sin\theta}{p + q\cos\theta}

The above formula is known as the Triangle Law of Vector Addition.

Parallelogram Law of Vector Addition

This law is just another way of understanding vector addition. This law states that if two vectors acting on the same point are represented by the sides of the parallelogram, then the resultant vector of these vectors is represented by the diagonals of the parallelograms. The figure below shows these two vectors represented on the side of the parallelogram. 

Solved Examples on Scalar and Vector

Example 1: Find the magnitude of v = i + 4j. 

Solution: 

|v| = \sqrt{a^2 + b^2}

a = 1, b = 4

|v| = \sqrt{1^2 + 4^2}

|v| = \sqrt{1^2 + 4^2}

|v| = √17

Example 2: A vector is given by, v = i + 4j. Find the magnitude of the vector when it is scaled by a constant of 5. 

Solution: 

|v| = \sqrt{a^2 + b^2}

5|v| = |5v| 

a = 1, b = 4

|5v|

|5(i + 4j)| 

|5i + 20j| 

|v| = \sqrt{5^2 + 20^2}

|v| = \sqrt{25 + 400}

|v| = √425

Example 3: A vector is given by, v = i + j. Find the magnitude of the vector when it is scaled by a constant of 0.5. 

Solution: 

|v| = \sqrt{a^2 + b^2}

0.5|v| = |0.5v| 

a = 1, b = 1

|0.5v|

|0.5(i + j)| 

|0.5i + 0.5j| 

|v| = \sqrt{0.5^2 + 0.5^2}

|v| = \sqrt{0.25 + 0.25}

|v| = √0.5

Example 4: Two vectors with magnitude 3 and 4. These vectors have a 90° angle between them. Find the magnitude of the resultant vectors. 

Solution: 

Let the two vectors be given by p and q. Then resultant vector "r" is given by, 

|r| = \sqrt{|p|^2 + |q|^2 + 2|p||q|cos(\theta)}

|p| = 3, |q| = 4 and \theta = 90^o

|r| = \sqrt{|p|^2 + |q|^2 + 2|p||q|cos(\theta)}

|r| = \sqrt{|3|^2 + |4|^2 + 2|3||4|cos(90)}

|r| = \sqrt{|3|^2 + |4|^2}

|r| = \sqrt{9 + 16}

|r| = \sqrt{9 + 16}                    

|r| = 5

Example 5: Two vectors with magnitude 10 and 9. These vectors have a 60° angle between them. Find the magnitude of the resultant vectors. 

Solution: 

Let the two vectors be given by p and q. Then resultant vector "r" is given by, 

|r| = \sqrt{|p|^2 + |q|^2 + 2|p||q|cos(\theta)}

|p| = 10, |q| = 9 and \theta = 60^o

|r| = \sqrt{|p|^2 + |q|^2 + 2|p||q|cos(\theta)}

|r| = \sqrt{|10|^2 + |9|^2 + 2|10||9|cos(60)}

|r| = \sqrt{|10|^2 + |9|^2 + (10)(9)}

|r| = \sqrt{100 + 81 + 90}

|r| = \sqrt{271}   

Conclusion

A scalar is a physical quantity that has only magnitude (or size) and no direction, whereas a vector is a quantity that possesses both magnitude and direction. Scalars are used to describe quantities like temperature and mass, where direction is irrelevant. In contrast, vectors are essential for representing physical quantities like displacement and velocity, where direction plays a key role.

You may also Read,

• Vector Algebra
• Dot and Cross Product on Vectors