What are the types of vectors?

The main types of vectors include: Zero Vector Unit Vector Equal Vectors Position Vector Collinear Vectors Coplanar Vectors

What is the difference between a zero vector and a unit vector?

A zero vector has magnitude zero and no specific direction. A unit vector has a magnitude of 1 and indicates direction.

Why are position vectors important?

A position vector helps represent the location of a point in space with respect to the origin, useful in geometry and physics for displacement and motion problems.

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Types of Vectors

In Mathematics and Physics, vectors are quantities that have both magnitude and direction. They are essential in representing physical quantities like force, velocity, and displacement. Understanding the types of vectors is important for solving problems in mechanics, electromagnetism, and geometry.

1.0What Are Vectors?

A vector is represented by an arrow where:

The length of the arrow shows the magnitude.
The direction of the arrow shows the direction of the vector.

Vectors are usually denoted in bold (e.g., A, B) or with an arrow above the letter (e.g., \vec{A}, \vec{B} ).

2.0Types of Vectors

Vectors are classified based on their properties and applications. Below are the main types of vectors you should know:

1. Zero Vector

A vector with zero magnitude and no specific direction.
Denoted as:

$0 = 0$

Example: The displacement of a stationary object.

2. Unit Vector

A vector with magnitude equal to 1, used to specify direction.
Denoted as $\hat{i}, \hat{j}$ and $\hat{k}$ in 3D space.
Example:

$\hat{i} = (1, 0, 0), \hat{j} = (0, 1, 0), \hat{k} = (0, 0, 1)$

3. Equal Vectors

Vectors having the same magnitude and direction, regardless of their initial point.
Example:

$A = (3, 4), B = (3, 4) t h e n A = B$

4. Position Vector

A vector that represents the position of a point with respect to the origin.
Example: The position vector of point P(x, y, z) is:

$OP = x \hat{i} + y \hat{j} + z \hat{k}$

5. Zero Vector vs. Null Vector

A Zero Vector is the vector with magnitude zero.
A Null Vector is sometimes used interchangeably but typically refers to a zero vector in context.

6. Collinear Vectors

Vectors lying along the same line or parallel lines.
Example:

$A = (2, 4), B = (1, 2)$

Since $A = 2 B$ , are collinear vectors.

7. Coplanar Vectors

Vectors lying in the same plane.
Example: Vectors $A, B$ and $c$ are coplanar vectors if:

$A . (B \times C) = 0$

3.0Types of Vectors with Examples Summary Table

Type of Vector	Definition	Example
Zero Vector	Magnitude = 0, no direction	$0 = 0$
Unit Vector	Magnitude = 1, specifies direction	$\hat{i}, \hat{j}, \hat{k}$
Equal Vectors	Same magnitude & direction	$A = (3, 4), B = (3, 4)$
Position Vector	Vector from origin to a point (x, y, z)	$OP = x \hat{i} + y \hat{j} + z \hat{k}$
Collinear Vectors	Lie along same or parallel lines	$A = 2 B$
Coplanar Vectors	Lie in the same plane	$A . (B \times C) = 0$

4.0Solved Examples on Types of Vectors

Example 1: Given $A = 3 \hat{i} + 4 \hat{j}$ and $B = - 6 \hat{i} - 8 \hat{j}$ check whether $A$ and $B$ are collinear.

Solution:

Two vectors are collinear if one is a scalar multiple of the other.
Here,

$B = - 2 A$

Thus, $A an d B$ are collinear.

Example 2: Find the position vector of point P(2, -3, 5).

Solution:

The position vector \vec{OP} is:

$OP = 2 \hat{i} - 3 \hat{j} + 5 \hat{k}$

So,

$OP = 2 \hat{i} - 3 \hat{j} + 5 \hat{k}$

Example 3: Determine whether the vectors $A = 2 \hat{i} + \hat{j} + \hat{k}, B = \hat{i} + 2 \hat{j} + 3 \hat{k}$ and $C = 3 \hat{i} + 5 \hat{j} + 7 \hat{k}$ are coplanar.

Solution:

Vectors are coplanar if:

$A . (B \times C) = 0$

First, compute $B \times C$ :

$B \times C = \hat{i} 13 \hat{j} 25 \hat{k} 37$

$= \hat{i} (2 \cdot 7 - 3 \cdot 5) - \hat{j} (1 \cdot 7 - 3 \cdot 3) + \hat{k} (1 \cdot 5 - 2 \cdot 3)$

$= \hat{i} (14 - 15) - \hat{j} (7 - 9) + \hat{k} (5 - 6)$

$= - \hat{i} + 2 \hat{j} - \hat{k}$

Now compute $A . (B \times C)$ :

$(2 \hat{i} + \hat{j} + \hat{k}) \cdot (- \hat{i} + 2 \hat{j} - \hat{k})$

$= (2) (- 1) + (1) (2) + (1) (- 1)$

$= - 2 + 2 - 1 = - 1$

Since $A \cdot (B \times C) \neq = 0$ _, are not coplanar vectors.

Example 4: Find the unit vector in the direction of $A = 3 \hat{i} - 4 \hat{j} + 12 \hat{k}$

Solution:
Step 1: Find the magnitude of $A$ :

$∣ A ∣ = 3^{2} + (- 4)^{2} + 1 2^{2} = 9 + 16 + 144 = 169 = 13$

Step 2: Unit vector $\hat{A}$ in the direction of $A$ :

$\hat{A} = \frac{A}{∣ A ∣} = \frac{3 i ^ - 4 j ^ + 12 k ^}{13}$

Final Answer:

$\hat{A} = \frac{3}{13} \hat{i} - \frac{4}{13} \hat{j} + \frac{12}{13} \hat{k}$

Example 5: Given

$A = \hat{i} + 2 \hat{j} + 3 \hat{k}$ and $B = 4 \hat{i} - \hat{j} + 2 \hat{k}$ find $A \cdot B$

Solution:

Dot product formula:

$A \cdot B = (1) (4) + (2) (- 1) + (3) (2) = 4 - 2 + 6 = 8$

Final Answer: 8

Example 6: If

$A = 2 \hat{i} + 3 \hat{j} + λ \hat{k} an d v ec B = - \hat{i} + 4 \hat{j} + 2 \hat{k}$

are perpendicular, find \lambda.

Solution:

Perpendicular vectors:

$A \cdot B = 0$

Compute the dot product:

$(2) (- 1) + (3) (4) + (λ) (2) = - 2 + 12 + 2 λ = 0$

Simplify:

$10 + 2 λ = 0 ⟹ λ = - 5$

Final Answer: −5

Example 7: Check whether the following vectors are coplanar:

$A = \hat{i} + 2 \hat{j} + 3 \hat{k}, B = 2 \hat{i} + 4 \hat{j} + 6 \hat{k}, C = 3 \hat{i} + 6 \hat{j} + 9 \hat{k}$

Solution:

Vectors are coplanar if:

$A \cdot (B \times C) = 0$

First, compute $B \times C$ :

$B \times C = \hat{i} 23 \hat{j} 46 \hat{k} 69$

$= \hat{i} (4 \times 9 - 6 \times 6) - \hat{j} (2 \times 9 - 6 \times 3) + \hat{k} (2 \times 6 - 4 \times 3)$

$= \hat{i} (36 - 36) - \hat{j} (18 - 18) + \hat{k} (12 - 12) = 0$

Thus,

$A \cdot (B \times C) = A \cdot 0 = 0$

So, the vectors are coplanar.

5.0Practice Questions on Types of Vectors

Given $A = \hat{i} + 2 \hat{j} + 3 \hat{k}$ and $B = 2 \hat{i} + 4 \hat{j} + 6 \hat{k}$ determine if $A$ and $B$ are collinear.
Find the position vector of point Q(-1, 4, 2).
Are the vectors $P = 3 \hat{i} - \hat{j} + 2 \hat{k}, Q = - \hat{i} + 2 \hat{j} - \hat{k}, an d R = 2 \hat{i} + \hat{j} + \hat{k}$ coplanar?
If $A = 4 \hat{i} + 3 \hat{j}$ and $B = - 2 \hat{i} - 1.5 \hat{j}$ find the scalar multiple relationship between them.

Also Read:

Vector Algebra	Components of a Vector	Dot Product of Two Vectors Questions
Multiplication of Vector	Vector Triple Product	Angle Between Two Vectors
Eigenvectors of a Matrix	Dot Product of Two Vectors	Vector Algebra PYQs

Join ALLEN!

(Session 2026 - 27)

Name

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Class

Choose class

Goal

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Preferred Programs

Preferred Mode

State

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I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Types of Vectors

In Mathematics and Physics, vectors are quantities that have both magnitude and direction. They are essential in representing physical quantities like force, velocity, and displacement. Understanding the types of vectors is important for solving problems in mechanics, electromagnetism, and geometry.

1.0What Are Vectors?

A vector is represented by an arrow where:

The length of the arrow shows the magnitude.
The direction of the arrow shows the direction of the vector.

Vectors are usually denoted in bold (e.g., A, B) or with an arrow above the letter (e.g., \vec{A}, \vec{B} ).

2.0Types of Vectors

Vectors are classified based on their properties and applications. Below are the main types of vectors you should know:

1. Zero Vector

A vector with zero magnitude and no specific direction.
Denoted as:

$0 = 0$

Example: The displacement of a stationary object.

2. Unit Vector

A vector with magnitude equal to 1, used to specify direction.
Denoted as $\hat{i}, \hat{j}$ and $\hat{k}$ in 3D space.
Example:

$\hat{i} = (1, 0, 0), \hat{j} = (0, 1, 0), \hat{k} = (0, 0, 1)$

3. Equal Vectors

Vectors having the same magnitude and direction, regardless of their initial point.
Example:

$A = (3, 4), B = (3, 4) t h e n A = B$

4. Position Vector

A vector that represents the position of a point with respect to the origin.
Example: The position vector of point P(x, y, z) is:

$OP = x \hat{i} + y \hat{j} + z \hat{k}$

5. Zero Vector vs. Null Vector

A Zero Vector is the vector with magnitude zero.
A Null Vector is sometimes used interchangeably but typically refers to a zero vector in context.

6. Collinear Vectors

Vectors lying along the same line or parallel lines.
Example:

$A = (2, 4), B = (1, 2)$

Since $A = 2 B$ , are collinear vectors.

7. Coplanar Vectors

Vectors lying in the same plane.
Example: Vectors $A, B$ and $c$ are coplanar vectors if:

$A . (B \times C) = 0$

3.0Types of Vectors with Examples Summary Table

Type of Vector	Definition	Example
Zero Vector	Magnitude = 0, no direction	$0 = 0$
Unit Vector	Magnitude = 1, specifies direction	$\hat{i}, \hat{j}, \hat{k}$
Equal Vectors	Same magnitude & direction	$A = (3, 4), B = (3, 4)$
Position Vector	Vector from origin to a point (x, y, z)	$OP = x \hat{i} + y \hat{j} + z \hat{k}$
Collinear Vectors	Lie along same or parallel lines	$A = 2 B$
Coplanar Vectors	Lie in the same plane	$A . (B \times C) = 0$

4.0Solved Examples on Types of Vectors

Example 1: Given $A = 3 \hat{i} + 4 \hat{j}$ and $B = - 6 \hat{i} - 8 \hat{j}$ check whether $A$ and $B$ are collinear.

Solution:

Two vectors are collinear if one is a scalar multiple of the other.
Here,

$B = - 2 A$

Thus, $A an d B$ are collinear.

Example 2: Find the position vector of point P(2, -3, 5).

Solution:

The position vector \vec{OP} is:

$OP = 2 \hat{i} - 3 \hat{j} + 5 \hat{k}$

So,

$OP = 2 \hat{i} - 3 \hat{j} + 5 \hat{k}$

Example 3: Determine whether the vectors $A = 2 \hat{i} + \hat{j} + \hat{k}, B = \hat{i} + 2 \hat{j} + 3 \hat{k}$ and $C = 3 \hat{i} + 5 \hat{j} + 7 \hat{k}$ are coplanar.

Solution:

Vectors are coplanar if:

$A . (B \times C) = 0$

First, compute $B \times C$ :

$B \times C = \hat{i} 13 \hat{j} 25 \hat{k} 37$

$= \hat{i} (2 \cdot 7 - 3 \cdot 5) - \hat{j} (1 \cdot 7 - 3 \cdot 3) + \hat{k} (1 \cdot 5 - 2 \cdot 3)$

$= \hat{i} (14 - 15) - \hat{j} (7 - 9) + \hat{k} (5 - 6)$

$= - \hat{i} + 2 \hat{j} - \hat{k}$

Now compute $A . (B \times C)$ :

$(2 \hat{i} + \hat{j} + \hat{k}) \cdot (- \hat{i} + 2 \hat{j} - \hat{k})$

$= (2) (- 1) + (1) (2) + (1) (- 1)$

$= - 2 + 2 - 1 = - 1$

Since $A \cdot (B \times C) \neq = 0$ _, are not coplanar vectors.

Example 4: Find the unit vector in the direction of $A = 3 \hat{i} - 4 \hat{j} + 12 \hat{k}$

Solution:
Step 1: Find the magnitude of $A$ :

$∣ A ∣ = 3^{2} + (- 4)^{2} + 1 2^{2} = 9 + 16 + 144 = 169 = 13$

Step 2: Unit vector $\hat{A}$ in the direction of $A$ :

$\hat{A} = \frac{A}{∣ A ∣} = \frac{3 i ^ - 4 j ^ + 12 k ^}{13}$

Final Answer:

$\hat{A} = \frac{3}{13} \hat{i} - \frac{4}{13} \hat{j} + \frac{12}{13} \hat{k}$

Example 5: Given

$A = \hat{i} + 2 \hat{j} + 3 \hat{k}$ and $B = 4 \hat{i} - \hat{j} + 2 \hat{k}$ find $A \cdot B$

Solution:

Dot product formula:

$A \cdot B = (1) (4) + (2) (- 1) + (3) (2) = 4 - 2 + 6 = 8$

Final Answer: 8

Example 6: If

$A = 2 \hat{i} + 3 \hat{j} + λ \hat{k} an d v ec B = - \hat{i} + 4 \hat{j} + 2 \hat{k}$

are perpendicular, find \lambda.

Solution:

Perpendicular vectors:

$A \cdot B = 0$

Compute the dot product:

$(2) (- 1) + (3) (4) + (λ) (2) = - 2 + 12 + 2 λ = 0$

Simplify:

$10 + 2 λ = 0 ⟹ λ = - 5$

Final Answer: −5

Example 7: Check whether the following vectors are coplanar:

$A = \hat{i} + 2 \hat{j} + 3 \hat{k}, B = 2 \hat{i} + 4 \hat{j} + 6 \hat{k}, C = 3 \hat{i} + 6 \hat{j} + 9 \hat{k}$

Solution:

Vectors are coplanar if:

$A \cdot (B \times C) = 0$

First, compute $B \times C$ :

$B \times C = \hat{i} 23 \hat{j} 46 \hat{k} 69$

$= \hat{i} (4 \times 9 - 6 \times 6) - \hat{j} (2 \times 9 - 6 \times 3) + \hat{k} (2 \times 6 - 4 \times 3)$

$= \hat{i} (36 - 36) - \hat{j} (18 - 18) + \hat{k} (12 - 12) = 0$

Thus,

$A \cdot (B \times C) = A \cdot 0 = 0$

So, the vectors are coplanar.

5.0Practice Questions on Types of Vectors

Given $A = \hat{i} + 2 \hat{j} + 3 \hat{k}$ and $B = 2 \hat{i} + 4 \hat{j} + 6 \hat{k}$ determine if $A$ and $B$ are collinear.
Find the position vector of point Q(-1, 4, 2).
Are the vectors $P = 3 \hat{i} - \hat{j} + 2 \hat{k}, Q = - \hat{i} + 2 \hat{j} - \hat{k}, an d R = 2 \hat{i} + \hat{j} + \hat{k}$ coplanar?
If $A = 4 \hat{i} + 3 \hat{j}$ and $B = - 2 \hat{i} - 1.5 \hat{j}$ find the scalar multiple relationship between them.

Also Read:

Vector Algebra	Components of a Vector	Dot Product of Two Vectors Questions
Multiplication of Vector	Vector Triple Product	Angle Between Two Vectors
Eigenvectors of a Matrix	Dot Product of Two Vectors	Vector Algebra PYQs

Frequently Asked Questions

What are the types of vectors?

What is the difference between a zero vector and a unit vector?

Why are position vectors important?

Frequently Asked Questions

What are the types of vectors?

What is the difference between a zero vector and a unit vector?

Why are position vectors important?

Join ALLEN!

Types of Vectors

1.0What Are Vectors?

2.0Types of Vectors

3.0Types of Vectors with Examples Summary Table

4.0Solved Examples on Types of Vectors

5.0Practice Questions on Types of Vectors

Table of Contents

Join ALLEN!

Types of Vectors

1.0What Are Vectors?

2.0Types of Vectors

3.0Types of Vectors with Examples Summary Table

4.0Solved Examples on Types of Vectors

5.0Practice Questions on Types of Vectors

Table of Contents