Types of Vectors in Maths with Examples and Diagrams
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.
FAQs on Types of Vectors: Definitions, Properties, and Examples
1. What are the main types of vectors in Maths?
The main types of vectors in mathematics include: Zero vectors (magnitude 0, no specific direction); Unit vectors (magnitude 1); Position vectors (representing a point's location relative to an origin); Equal vectors (same magnitude and direction); Like vectors (same direction); Unlike vectors (opposite directions); Co-initial vectors (same starting point); Collinear vectors (parallel to the same line); Coplanar vectors (lying in the same plane); and Displacement vectors (representing change in position).
2. What is the fundamental difference between a scalar and a vector quantity?
Scalars are quantities defined solely by their magnitude (a numerical value), such as mass, temperature, or speed. Vectors, however, require both magnitude and direction for complete description; examples include force, velocity, and displacement.
3. How can two vectors be equal if they start from different points?
Two vectors are considered equal if they possess the same magnitude and direction. Their starting points are irrelevant to their equality; only the length and orientation matter. Think of it as representing the same displacement, regardless of the starting location.
4. Why is the Zero Vector considered a vector if it has no specific direction?
The zero vector (or null vector) is essential for maintaining consistency in vector algebra. Although it has zero magnitude and an undefined direction, it serves as the additive identity, similar to zero in regular arithmetic. For instance, adding a vector to its negative results in the zero vector (a + (-a) = 0).
5. What is the practical importance of a Unit Vector?
A unit vector is crucial because it represents pure direction with a magnitude of one. Its primary use is to define the direction of any other vector. Multiplying a scalar (magnitude) by a unit vector creates a vector of any length in that specified direction. The standard basis vectors (î, ĵ, k̂) are unit vectors defining the x, y, and z axes.
6. How do you distinguish between Collinear and Co-initial vectors?
Co-initial vectors share the same starting point, irrespective of magnitude or direction. Collinear vectors are parallel to the same line, regardless of their starting points; their magnitudes and directions can differ. The key is orientation for collinear vectors and starting point for co-initial vectors.
7. What are some real-world examples of vector quantities?
Real-world vector quantities abound: Force (pushing a box requires both strength and direction); Velocity (a car's speed and direction); Displacement (change in position); and Acceleration (rate of change of velocity).
8. How does a position vector help describe an object's location?
A position vector provides a unique spatial address for a point relative to a fixed origin (O). The vector (OP) has a magnitude equal to the distance from the origin to the point and points from the origin to the point. In a coordinate system, its components directly represent the point's coordinates.
9. What are like and unlike vectors?
Like vectors point in the same direction, while unlike vectors point in opposite directions. The magnitudes can differ; the critical factor is their orientation relative to each other.
10. What are coplanar vectors?
Coplanar vectors are three or more vectors that all lie within the same plane in three-dimensional space.
11. What is a displacement vector?
A displacement vector represents the change in position of a point from an initial location to a final location. It describes both the distance and direction of the movement.
12. What is the negative of a vector?
The negative of a vector has the same magnitude but the opposite direction. If vector 'a' is represented by an arrow, then vector '-a' would be an arrow of the same length but pointing in the precisely reverse direction.
