Scalar And Vector

Mathematics and Science were invented by humans to understand and describe the world around us. A lot of mathematical quantities are used in Physics to explain the concepts clearly. A few examples of these include force, speed, velocity and work. These quantities are often described as being a scalar or a vector quantity. Scalars and vectors are differentiated depending on their definition. A scalar quantity is defined as the physical quantity that has only magnitude, for example, mass and electric charge. On the other hand, a vector quantity is defined as the physical quantity that has both, magnitude as well as direction like force and weight. The other way of differentiating these two quantities is by using a notation.

Scalar and Vector

What Is Scalar Quantity?

Scalar quantity is defined as

The physical quantity with magnitude and no direction.

Some physical quantities can be described just by their numerical value (with their respective units) without directions (they don’t have any direction). The addition of these physical quantities follows the simple rules of the algebra. That is only their magnitudes are added.

What is Scalar quantity example?

There are plenty of scalar quantity examples, some of the common examples are:

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

What Is Vector Quantity?

A vector quantity is defined as

The physical quantity that has both direction as well as magnitude.

A vector with the value of magnitude equal to one and direction is called unit vector represented by a lowercase alphabet with a “hat” circumflex. That is “û“.

What are some examples of vector quantities?

Vector quantity examples are many, some of them are given below:

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

Difference Between Scalar and Vector

The difference between Scalar and Vector is crucial to understand in physics learning. Below are a few differences for better understanding.

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

The addition and subtraction of vector quantities does 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:

  • Addition of vectors means finding the resultant of a number of vectors acting on a body.
  • The component vectors whose resultant is to be calculated are independent of each other. Each vector acts as if the other vectors were absent.
  • Vectors can be added geometrically but not algebraically.
  • Vector addition is commutative in nature, i.e., \(\underset{A}{\rightarrow}+\underset{B}{\rightarrow}=\underset{B}{\rightarrow}+\underset{A}{\rightarrow}\)

Now, talking 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 \(\underset{A}{\rightarrow}\) and \(\underset{B}{\rightarrow}\) as shown in the figure below. We required to subtract \(\underset{B}{\rightarrow}\) from \(\underset{A}{\rightarrow}\). It is just the same as adding \(\underset{-B}{\rightarrow}\) and \(\underset{A}{\rightarrow}\). The resultant is shown in the figure below
Subtraction of Vectors
Resultant Vector

Vector Notation

For vector quantity usually, an arrow is used on the top like \(\underset{v}{\rightarrow}\) which represents the vector value of the velocity and also explains that the quantity has both magnitudes as well as direction.

Following is the table explaining other related concepts:

Scalar And Vector Quantities Problems With Solutions

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 either be positive or negative along an axis.

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

\(\sum y=10\,m-5\,m=5\,m\)

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 movement towards the west to be negative.

\(\sum y=-3\,m+12\,m=9\,m\)

He moved a net of 9 m to the east.

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

\(D^2=(\sum x^2)+(\sum y^2)\)

Substituting the values, we get

\(D^2=(9^2)+(5^2)\) \(D^2=(106)^2\) \(\sqrt{D^2}=\sqrt{(106)^2}\) \(D=10.30\,m\)

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.

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