Table of Content

• Scalar Quantities Definition
• Vector Quantities 
• Vector Notation
• Scalar and Vector Quantity
• Equality of Vectors
• Multiplication of Vectors with Scalar
• Addition of Vectors
• Triangle Law of Vector Addition
• Parallelogram Law of Vector Addition
• Examples on Scalar and Vector

Difference Between Scalar and Vector Quantity
Scalar	Vector
Scalar quantities have magnitude or size only.	Vector quantities have both magnitude and direction.
It is known that every scalar exists in one dimension only.	Vector quantities can exist in one, two, or three-dimension.
Whenever there is a change in a scalar quantity, can correspond to a change in its magnitude also.	Any change in a vector quantity can correspond to cha change in either its magnitude or direction or both.
These quantities can not be resolved into their components.	These quantities can be resolved into their components, using the sine or cosine of the adjacent angle.
Any mathematical process that involves more than two scalar quantities will only give scalars.	Mathematical operations on two or more vectors can provide either a scalar or a vector as a result. For instance, the dot product of two vectors only produces a scalar, whereas the cross product, sum, or subtraction of two vectors gives a vector.
Some examples of Scalar quantities are: Mass Speed Distance Time Area Volume	Some examples of Vector quantities are: Velocity Force Pressure Displacement Acceleration

Equality of Vectors

Two vectors are considered to be equal when they have the same magnitude and 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 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, 

[Tex]|k\vec{v}| = k|\vec{v}|[/Tex] 

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. 

[Tex]\vec{a} + \vec{b} = \vec{c} [/Tex]

[Tex]\vec{a} + \vec{b} = \vec{b} + \vec{a}     [/Tex]  – (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, 

[Tex]\vec{p} + \vec{q} [/Tex]

The magnitude of the resultant vector is given by, 

[Tex]\sqrt{|p|^2 + |q|^2 + 2|p||q|cos(\theta)} [/Tex]

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

[Tex]tan (\phi) = \dfrac{q\sin\theta}{p + q\cos\theta} [/Tex]

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. 

Also, Check:

• Vector Algebra
• Dot and Cross Product of Vectors

Examples on Scalar and Vector

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

Solution: 

|v| = [Tex]\sqrt{a^2 + b^2} [/Tex]

a = 1, b = 4

|v| = [Tex]\sqrt{1^2 + 4^2} [/Tex]

|v| = [Tex]\sqrt{1^2 + 4^2} [/Tex]

|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| = [Tex]\sqrt{a^2 + b^2} [/Tex]

5|v| = |5v| 

a = 1, b = 4

|5v|

|5(i + 4j)| 

|5i + 20j| 

|v| = [Tex]\sqrt{5^2 + 20^2} [/Tex]

|v| = [Tex]\sqrt{25 + 400} [/Tex]

|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| = [Tex]\sqrt{a^2 + b^2} [/Tex]

0.5|v| = |0.5v| 

a = 1, b = 1

|0.5v|

|0.5(i + j)| 

|0.5i + 0.5j| 

|v| = [Tex]\sqrt{0.5^2 + 0.5^2} [/Tex]

|v| = [Tex]\sqrt{0.25 + 0.25} [/Tex]

|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, 

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

|p| = 3, |q| = 4 and [Tex]\theta = 90^o [/Tex]

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

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

[Tex]|r| = \sqrt{|3|^2 + |4|^2} [/Tex]

[Tex]|r| = \sqrt{9 + 16} [/Tex]

[Tex]|r| = \sqrt{9 + 16}                    [/Tex] 

|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, 

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

|p| = 10, |q| = 9 and [Tex]\theta = 60^o [/Tex]

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

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

[Tex]|r| = \sqrt{|10|^2 + |9|^2 + (10)(9)} [/Tex]

[Tex]|r| = \sqrt{100 + 81 + 90} [/Tex]

[Tex]|r| = \sqrt{271}   [/Tex] 

Scalars and Vectors-FAQs

What do you mean by Scalars and Vectors, in physics?

Scalars are the physical quantities that have magnitude or size only. While vectors are the physical quantities that have both magnitude and direction.

What are examples of Vectors Quantities?

Here are some important examples of vectors quantites are:

• Velocity
• Force
• Pressure
• Displacement
• Acceleration
• Thrust

What are some Scalar Quantities?

Here are some important examples of scalars are:

• Mass
• Speed
• Distance
• Time
• Area
• Volume

Is Force is a Scalar or a Vector Quantity?

Since force is a physical quantity that has both magnitude and direction. Therefore, it’s a vector quantity.

What is Difference Between Distance and Displacement?

The main difference between distance and displacement is that the distance has magnitude only and is a scalar quantity. However, displacement has both magnitude and direction so it is a vector quantity.