Vectors – Scalar Quantity, Vector Quantity with Examples

Vectors

In mechanics (motion, force, etc.), we come across various measurable, derivable, and formulating quantities. To measure or calculate a particular set of quantities, you have to understand which category they belong to. In this session, we will discuss and understand those quantities by which we can calculate the motion.

Mainly we consider a quantity as

1. Scalars
2. Vectors

Explanation

Scalar Quantity

Scalars are physical quantities that have only magnitude but not direction.

Scalars have only value but not direction.

For example, a car is moving at a speed of 35m/s. Here only the value of the speed is mentioned as 35m/s, but it is not said in which direction the car is moving.

Examples of Scalar Quantity

1. Speed
2. Distance

Vector Quantity

A vector that has both directions as well as magnitude.

Example: A car is moving with a velocity of 35 m/s in the east direction.

Here unlike the scalar, the value and the direction are mentioned.

If we consider coordinates such as the X-axis and Y-axis. Usually, the direction is mentioned as positive or negative X-axis and positive or negative Y-axis.

Examples of Vector Quantity

• Linear momentum
• Acceleration
• Displacement

General Notation of Vectors

Generally, vectors are represented by capital letters such as P, Q, R, etc.

The symbol “–>” is called direction and is used above the vector-like P-> to represent the vector.

Suppose a vector is represented like |P|; only the magnitude is represented on the vector. There is also a graphical representation of vectors, like the image below.

In the above image, the red line represents the vector drawn in a plane (X and Y-axis or
2-dimensions); observe the symbol ϴ in the graph.

What does it represent?

The Direction of the Vector

That symbol ϴ represents the direction of the vector, and it is crucial to understand the direction of the vector as vectors have both magnitude and direction.

Representing a Vector

• A vector is always represented with direction.
• Draw a vector with the head and the tail, where the head represents the direction of the vector.
• Vectors are represented on straight lines; only they do not have a curve or circular representation.

Types of Vectors

• Unit vector:  A vector whose magnitude is UNIT or ONE is called a unit vector; generally, unit vectors are represented by the symbol “^,” called CAP.
• Zero vector: A vector whose magnitude is ZERO or NULL is called a zero vector.
• Parallel vectors: Vectors whose directions are the same are called parallel vectors.
• Anti-parallel vectors: Vectors whose directions are opposite or different are called anti-parallel vectors.
• Equal vectors: Vectors whose magnitudes are the same are called similar vectors.
• Coplanar vectors: Vectors in the same plane are called coplanar vectors.
• Concurrent vectors: Vectors whose starting or initial point is the same are called concurrent vectors.

Angle Between the Vectors

Let us consider two vectors, P & Q; they are concurrent vectors; let us say the angle between them is ϴ; therefore, we must understand that the angles between the vectors are up to 0 to 180 only.

Multiplication of Vectors by Scalars

Let A be a vector and consider m as a scalar; if the vector is multiplied by the scalar, it can be written as |mA|=mA (m may be positive or negative).

Division of Vectors by Scalars

Let A be a vector and consider m a scalar; if the vector is divided by the scalar, it can be written as 1/m x A=|A/m|=A/m (m may be positive or negative).

Addition Of Vectors

As we know, vectors are measurable quantities with magnitude and direction; they can be added, subtracted, multiplied, and divided.

Remember, vectors also have direction, so we cannot add them as we add the numerical. Therefore, there exists a separate rule and pattern in adding the vectors.

Triangle Law

Let us consider two vectors, A and B.

And the vectors are represented like the image above.

Note: Triangle law is applicable when a head and the tail of two vectors are connected.

Then the resultant vector ‘R’ after adding the vectors A and B is given by,

R = A + B

If the vectors are orthogonal,

then R²= A²+B²

Direction: Tanϴ= |B|/|A|

Parallelogram Law of Vector Addition

Let us consider two vectors, P and Q.

Note: Parallelogram law is used when the vectors are added with their tails, like in the image above.

Let the two vectors be represented in magnitude and direction by two adjacent sides of parallelogram OPQS, drawn from a point O.

According to the parallelogram law of vectors, their resultant vector will be represented by a diagonal.

of the parallelogram.

Magnitude of R:

Draw QN perpendicular to OP produced.

From the figure, OP = A, OS = b, OQ = R and ∠SOP = ∠QPN = θ

△QNP, PN = PQ cosθ = B cosθ

QN = PQ sinθ = B sinθ

In the right-angled triangle, ONQ, we have

OQ² = ON² + NQ²

= (OP+ PN)² + NQ²

Or R² = (A+B cos θ)² + (B sinθ)²

Or R² = A² + 2AB cos θ + B² (cos²θ + sin²θ)

Or R² = A² + 2AB cos θ + B²

Or

Direction

Tanϴ = (B sinθ/A + B cosθ)