Types of Vectors: Definitions, Properties, and Examples

The concept of types of vectors plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding various types of vectors forms the bedrock for topics like vector algebra, physics mechanics, engineering, and even computer science. In this guide, we’ll help you quickly master the definitions, properties, and key differences between the main types of vectors in maths.

What Are Types of Vectors?

Types of vectors refers to the classification of vectors in mathematics based on their magnitude, direction, position, and relation to other vectors. A vector is a quantity that has both magnitude (size) and direction, such as force, velocity, or displacement. You’ll find this concept applied in areas such as Physics (forces and velocity), Engineering (directional movement), and Biology (gene vectors).

Key Types of Vectors in Maths

• Zero (Null) Vector: Magnitude is zero; direction is undefined. Example: (0, 0, 0)
• Unit Vector: Magnitude is exactly 1. Example: \( \hat{i}, \hat{j}, \hat{k} \) in 3D
• Position Vector: Represents the position of a point from the origin. Example: \( \vec{OP} \)
• Equal Vectors: Same magnitude and direction, regardless of position. Example: \( \vec{AB} = \vec{CD} \)
• Opposite Vectors: Same magnitude, opposite direction. Example: \( \vec{a} \) and \( -\vec{a} \)
• Like Vectors: Parallel and same direction
• Unlike Vectors: Parallel but opposite direction
• Collinear (Parallel) Vectors: Lie along the same or parallel lines
• Coplanar Vectors: Lie on the same plane
• Co-initial Vectors: Same initial point
• Displacement Vector: Represents the change from one position to another
• Negative Vector: Same magnitude as given vector but in the opposite direction

Table: Types of Vectors and Their Properties

Vector Type	Symbol / Representation	Key Property	Example
Zero/Null Vector	\( \vec{0} \)	Magnitude = 0; no direction	(0,0) or (0,0,0)
Unit Vector	\( \hat{a} \)	Magnitude = 1; shows direction	\( \hat{i}, \hat{j}, \hat{k} \)
Position Vector	\( \vec{OP} \)	Points from origin to P	(x, y, z)
Equal Vectors	\( \vec{a} = \vec{b} \)	Same magnitude, same direction	\( \vec{AB}, \vec{CD} \)
Opposite Vectors	\( \vec{a}, -\vec{a} \)	Same magnitude, opposite direction	North/South movement
Like Vectors	Parallel	Same direction, not necessarily same length	All eastward arrows
Unlike Vectors	Parallel, opposite	Opposite direction	Eastward & westward arrows
Collinear Vectors	Lines ||	Along same/parallel line	Force on same rope
Coplanar Vectors	In plane	All in the same flat surface	Vectors in paper
Co-initial Vectors	Same start point	Originates at same point	\( \vec{OA}, \vec{OB} \)
Displacement Vector	\( \vec{AB} \)	Shows change in position	Path from A to B

Memory Tip: Mnemonic for Types of Vectors

A quick way to remember the most common types of vectors: “ZUP CLED COCD”

• Z – Zero/null
• U – Unit
• P – Position
• C – Co-initial
• L – Like/Unlike
• E – Equal
• D – Displacement
• C – Collinear
• O – Opposite
• C – Coplanar
• D – Negative (Direction reversed)

Draw labeled arrows starting from a common point and show some going in equal, opposite, or parallel ways to visualize each vector type.

Real-Life & Cross-Subject Examples

Types of vectors are not only useful in Maths but also play an important role in:

• Physics: Force, velocity, acceleration, displacement — all are vectors.
• Biology: “Vectors” describe carriers of genes in biotechnology.
• Daily Life: Walking 5m north (displacement), wind blowing east (velocity).
• Engineering/Robotics: Position and movement of machine arms.

Students preparing for competitive exams (like JEE, NEET) will often encounter vector type identification in both Maths and Physics papers.

Solved Problems: Types of Vectors

Let’s practice how to identify and distinguish types of vectors for exams:

Example 1: Identify the vector

Given: \( \vec{AB} = (3, 4) \) and \( \vec{CD} = (3, 4) \)

1. Compare \( \vec{AB} \) and \( \vec{CD} \): Both have same direction and length.

2. Therefore: They are equal vectors.

Example 2: Is (0, 0) a zero vector?

1. Check if magnitude = 0: Yes, both components are zero.

2. Thus, (0, 0) is a zero or null vector.

Example 3: Vectors \( \vec{PQ} = (5,0) \) and \( \vec{RS} = (-5,0) \)

1. Magnitude is same: 5 units.

2. Directions are opposite.

3. Thus, they are opposite vectors.

Quiz: Test Your Understanding of Types of Vectors

• Which type of vector has magnitude zero?
• Give an example of unit vector in 3D.
• If two vectors are parallel and point the same way, what are they called?
• What’s the difference between co-initial and collinear vectors?

Quick Revision Table

Abbrev.	Vector Type	Property
ZV	Zero/Null Vector	Magnitude 0; no direction
UV	Unit Vector	Magnitude 1 unit
POSV	Position Vector	From origin to point
EV	Equal Vectors	Same length & direction
OPV	Opposite Vectors	Equal length, opposite direction
CIV	Co-initial Vectors	Same start point
CLV/CPV	Collinear/Coplanar Vectors	On same line/plane

Common Mistakes with Types of Vectors

• Assuming all vectors with same direction are equal (remember: length also matters).
• Confusing zero vector with unit vector.
• Mixing up co-initial and collinear vectors.

Related Concepts and Further Reading

Mastering types of vectors helps you solve problems in topics like Vector Algebra and understand the fundamental differences between scalars and vectors. Dive deeper for applications in real-world physics and engineering or learn more about special types like unit vectors and zero vectors.

We explored types of vectors—from definition, table summary, examples, common errors, and their connections to physics and daily life. Continue practicing with Vedantu to become confident in identifying and solving problems using each vector type, which is key for all major board exams and entrance tests.