Vectors: Definition, Types & Unit Vectors | Physics for JEE Main & Advanced PDF Download

Scalar and Vector Quantities

• In physics, we often deal with quantities like force, speed, velocity, and work. We classify these quantities as either scalar or vector. 
• A scalar quantity only has a magnitude, like mass or electric charge – it's just a number with no specific direction. 
• Vectors are mathematical entities used to represent quantities that have both magnitude and direction. They are commonly employed in various fields, including physics, engineering, computer science, and more. 

Scalar and Vector Quantities

Difference between Scalar and Vector Quantities

1. Scalar Quantity:

• Definition:. scalar quantity is defined by its magnitude alone, without any consideration of direction.
• Examples: Common examples of scalar quantities include mass, temperature, speed, and energy.
• Representation: Scalars are represented by a single numerical value accompanied by its unit, such as "5 kg" or "20 degrees Celsius."

2. Vector Quantity:

• Definition:. vector quantity is defined by both its magnitude and direction.
• Examples: Examples of vector quantities include displacement, velocity, acceleration, and force.
• Representation: Vectors are represented by a numerical value along with a specified direction. This is often depicted using an arrow that indicates the direction, such as "10 m North" or "25 m/s East."

Representation of a Vector

• Vectors are like arrows. Arrows have two important things: how long they are (that's their size), and the direction they point. 
• Think of a vector like an arrow, where one end is where it starts (that's the tail), and the other end is where it's pointing to (that's the head). 
• So, a vector has a length and shows you a specific direction, just like an arrow.

Examples of VectorsParts of a Vector

Types of Vectors

All these vectors are extremely important and the concepts are frequently required in mathematics and other higher-level science topics. The detailed explanations on each of these 10 vector types are given below.

1. Zero Vector

A zero vector is a vector when the magnitude of the vector is zero and the starting point of the vector coincides with the terminal point.

"In other words, for a vector the coordinates of the point A are the same as that of the point B then the vector is said to be a zero vector and is denoted by 0.

This follows that the magnitude of the zero vector is zero and the direction of such a vector is indeterminate.

2. Unit Vector

A vector which has a magnitude of unit length is called a unit vector.

Suppose if is a vector having a magnitude x then the unit vector is denoted by x̂ in the direction of the vector and has the magnitude equal to 1. Therefore,

It must be carefully noted that any two unit vectors must not be considered as equal, because they might have the same magnitude, but the direction in which the vectors are taken might be different.

3. Position Vector

If O is taken as reference origin and P is an arbitrary point in space then the vector is called as the position vector of the point.

Position vector simply denotes the position or location of a point in the three-dimensional Cartesian system with respect to a reference origin.

4. Co-initial Vectors

The vectors which have the same starting point are called co-initial vectors.

Co- initial Vectors

5. Like and Unlike Vectors

The vectors having the same direction are known as like vectors. On the contrary, the vectors having the opposite direction with respect to each other are termed to be unlike vectors.

6. Co-planar Vectors

Three or more vectors lying in the same plane or parallel to the same plane are known as co-planar vectors.

7. Collinear Vectors

Vectors that lie along the same line or parallel lines are known to be collinear vectors. They are also known as parallel vectors.

Two vectors are collinear if they are parallel to the same line irrespective of their magnitudes and direction. Thus, we can consider any two vectors as collinear vectors if and only if these two vectors are either along the same line or these vectors are parallel to each other in the same direction or opposite direction. For any two vectors to be parallel to one another, the condition is that one of the vectors should be a scalar multiple of another vector. 

The below figure shows the collinear vectors in the opposite direction.

8. Equal Vectors

Two or more vectors are said to be equal when their magnitude is equal and also their direction is the same.

The two vectors shown above, are equal vectors as they have both direction and magnitude equal.

9. Displacement Vector

If a point is displaced from position A to B then the displacement AB represents a vector which is known as the displacement vector.

10. Negative of a Vector

If two vectors are the same in magnitude but exactly opposite in direction then both the vectors are negative of each other.

For instance, if vector a has the same magnitude as vector b but points in the opposite direction, it can be represented as a = -b.

$Suppose\; if$

Magnitude of a Vector

The magnitude of a vector is a crucial measure that provides the numeric value for a given vector. It is denoted as |A| and can be calculated using the formula:

|A| = √(a² + b² + c²)

This formula summarizes the individual measures of the vector along the x, y, and z-axes.

Unit Vector: Representation & Notation

As we have studied the definition of unit vectors above, Let's study in detail more.

Unit Vectors î, ĵ, k̂

Unit Vector Notation and Formula

Unit vectors are denoted by the symbol (^) and can be calculated using the formula:

= A / |A|

Here, A represents the given vector, and |A| is the magnitude of vector A. The resulting unit vector has the same direction as A but a magnitude of 1.

To find a unit vector with the same direction as a given vector, simply divide the vector by its magnitude. For example, if vector v = (3, 4), its unit vector is calculated as follows:

|v| = √(3² + 4²) = 5

= v / |v| = (3, 4) / 5 = (3/5, 4/5)

Representation of  Unit Vectors

Vectors can be represented in both bracket and component formats. For a vector = (x, y, z), the unit vectorin bracket format is given by:

= (x/√(x² + y² + z²), y/√(x² + y² + z²), z/√(x² + y² + z²))

In component format, it is represented as:

= x/√(x² + y² + z²) . î + y/√(x² + y² + z²) . ĵ + z/√(x² + y² + z²) . k̂

Solved Example on Vectors

Example 1: Given the vectors: A = 3i + 2j – k and B = 5i +5j.
Determine:

1. Their magnitude.
2. The direction of B.
3. A + B
4. A -2 B
5. A unit vector parallel to A.
6. A vector of magnitude 2 and opposite to B

Ans.It is essential when working with vectors to use proper notation. Always draw an arrow over the letters representing vectors. You can also use bold characters to represent a vector quantity.
Vectors Aand Bare written using the unit vector notation.

1. The magnitude of Ais given by:

Similarly, the magnitude of Bis: 

The magnitude of a vector is always a positive number. 

2. The direction of B can be found in the following way

3. The vector sum of Aand Bis given by: 

4. We multiply B by -2 and then add A:

5. A unit vector is created from another one by dividing the last by its magnitude: 

6. We can find a vector of magnitude 2 and opposite to Bby multiplying a unit vector parallel to Bby -2: 

Example 2:  How many minimum number of coplanar vectors having different magnitudes can be added to give zero resultant?

Sol: According to the Triangle Law of vector addition, a minimum of three vectors is needed to get a zero resultant. Therefore, a minimum of 3 coplanar vectors is required to represent the same physical quantity with different magnitudes that can be added to give zero results.

Example3:The square of the resultant of two equal forces is three times their product. What is the angle between the forces?

Sol: Let A and B be the two forces and θ be the angle between them.

Also, A = B.

Given the square of the resultant of two equal forces is three times their product.

F ² _res = 3AB

⇒ F ² _res = A ² + B ² + 2AB cos θ

⇒ 3AB = A ² + B ² + 2AB cos θ

Since A = B,

⇒ 3A ² = 2A ² + 2A ² cos θ

⇒ A ² = 2A ² cos θ

⇒ cos θ = 1/2

⇒ θ = π/3

Example 4: Given A = 2i + 3j and B = i + j. The component of vector A along vector B is

Sol: The component of vector A along B is A.B / |B|
= (2i + 3j).(i + j)/√(1 + 1) 
= 5/√2

Example 5: Prove that the three vectors $<span\; style="background-color:\; transparent;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; color:\; black;"><strong\; style="word-break:\; break-word;\; font-weight:\; 700;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">are\; at\; right\; angles\; to\; one\; another.</strong></span>$

Sol. Let (all the three vectors are non-zero).

Thus the dot product of two non-zero vectors $\backslash mathbf\{a\}$a and b is zero, meaning the two vectors are perpendicular to each other.

Again,

and

As above, it follows that b and c are perpendicular and c and $\backslash mathbf\{a\}$a are perpendicular. Hence, all the three given vectors are perpendicular to each other.