Vector and Scalar-Definition, Vector Addition and Subtraction, Differences, Solved Problems

What Is a Scalar Quantity?

A scalar quantity is defined as the physical quantity with only magnitude and no direction. Such physical quantities can be described just by their numerical value without directions. The addition of these physical quantities follows the simple rules of algebra, and here, only their magnitudes are added.

Examples of Scalar Quantities

Some examples of scalar include:

• Mass
• Speed
• Distance
• Time
• Volume
• Density
• Temperature

What Is a Vector Quantity?

A vector quantity is defined as the physical quantity that has both directions as well as magnitude.
A vector with a value of magnitude equal to one is called a unit vector and is represented by a lowercase alphabet with a “hat” circumflex, i.e. “û“.

Examples of Vector Quantities

Examples of vector quantity include:

• Linear momentum
• Acceleration
• Displacement
• Momentum
• Angular velocity
• Force
• Electric field
• Polarization

Difference Between Scalars and Vectors

The differences between a scalar and vector in the table below:

	Vector	Scalar
Definition	A physical quantity with both the magnitude and direction.	A physical quantity with only magnitude.
Representation	A number (magnitude), direction using unit cap or arrow at the top and unit.	A number (magnitude) and unit
Symbol	Quantity symbol in bold and an arrow sign above	Quantity symbol
Direction	Yes	No
Example	Velocity and Acceleration	Mass and Temperature

Vector Addition and Subtraction

After understanding what is a vector, let’s learn vector addition and subtraction. The addition and subtraction of vector quantities do not follow the simple arithmetic rules. A special set of rules are followed for the addition and subtraction of vectors. Following are some points to be noted while adding vectors:

Now, about vector subtraction, it is the same as adding the negative of the vector to be subtracted. To better understand, let us look at the example given below.

Let us consider two vectors, A and B, as shown in the figure below. We need to subtract vector B from vector A. It is just the same as adding vector B and vector A. The resultant vector is shown in the figure below.

Vector Notation

For vector quantity usually, an arrow is used on the top as shown below, which represents the vector value of the velocity and also explains that the quantity has both magnitudes as well as direction.

\(\begin{array}{l}Vector\, Notation= \vec{v}\end{array} \)

Scalar and Vector Solved Problems

Q1: Given below is a list of quantities. Categorize each quantity as being either a vector or a scalar.

20 degrees Celsius

5 mi, North

256 bytes

5 m

30 m/sec, East

4000 Calories

Answer:

20 degrees Celsius	Scalar
5 mi, North	Vector
256 bytes	Scalar
5 m	Scalar
30 m/sec, East	Vector
4000 Calories	Scalar

Q2: Ashwin walks 10 m north, 12 m east, 3 m west and 5 m south and then stops to drink water. What is the magnitude of his displacement from his original point?

Answer: We know that displacement is a vector quantity; hence the direction Ashwin walks will be positive or negative along an axis.

To find the total distance travelled along the y-axis, let us consider the movement towards the north to be positive and the south to be negative.

\(\begin{array}{l}\sum y=10\,m-5\,m=5\,m\end{array} \)

He moved a net of 5 meters to the north along the y-axis.

Similarly, let us consider his movement towards the east to be positive and the west to be negative.

\(\begin{array}{l}\sum y=-3\,m+12\,m=9\,m\end{array} \)

He moved a net of 9 m to the east.

Using Pythagoras theorem, the resultant displacement can be found as follows:

\(\begin{array}{l}D^2=(\sum x^2)+(\sum y^2)\end{array} \)

Substituting the values, we get

\(\begin{array}{l}D^2=(9^2)+(5^2)\end{array} \)

\(\begin{array}{l}D^2=(106)^2\end{array} \)

\(\begin{array}{l}\sqrt{D^2}=\sqrt{(106)^2}\end{array} \)

\(\begin{array}{l}D=10.30\,m\end{array} \)

Q3. What is the magnitude of a unit vector?

Answer: The magnitude of a unit vector is unity. A unit vector has no units or dimensions.

Frequently Asked Questions – FAQS

Q1

What is vector and scalar quantity in Physics?

A scalar quantity is defined as the physical quantity that has only magnitude. On the other hand, a vector quantity is defined as the physical quantity that has both magnitude as well as direction.

Q2

How are vector and scalar different?

Vectors have both magnitude and direction but scalars have only magnitude.

Q3

How are vectors and scalars quantities alike?

Scalars and vectors both have specific unit and dimension. Both of these quantities are measurable. Moreover, both possess magnitude.

Q4

What are the examples of scalar?

Mass and electric charge are examples of scalars.

Q5

What are the examples of vectors?

Displacement and angular velocity are examples of vectors.