Vectors are specified as an object including magnitude plus direction. It depicts the movement of the object from one location to another. Vectors are named individually based on their features such as magnitude, direction, and their association with the different types of vectors. These different types of vectors help implement various arithmetic operations and calculations concerning vectors. If we consider vectors as a straight line then the extent of the line denotes the magnitude and the pointer on this line is the direction in which the vector is traveling as shown below:

Types of Vectors

There are 12 types of vectors that are as follows:

1. Zero Vector
2. Unit Vector
3. Position Vector
4. Equal Vector
5. Negative Vector
6. Collinear Vector
7. Co-initial Vector
8. Like and Unlike Vectors
9. Co-planar Vector
10. Displacement Vector
11. Orthogonal Vector
12. Concurrent vector

1. What is a Zero Vector?

A zero vector is a vector with zero magnitude. This implies that the starting point of the vector matches the final point. Such a vector has zero (0) magnitude and is denoted by 0. Consider an example of zero vector to understand the same.

If for a vector say \(\vec{PQ}\), the coordinates of the point P lie at the same position as that of the point Q then the vector is declared to be a zero vector.

This reflects that the magnitude of the zero vector is always zero plus the direction for such a vector is indeterminate. Also, the vector does not aim in any direction. The zero vectors are known as null vectors.

2. What is a Unit Vector?

A unit vector is a vector whose magnitude is of unit length. If \(\vec{x}\) is a vector whose magnitude is x, then unit vector of \(\vec{x}\) in the direction of x is denoted by \(\widehat{x}\) and is defined as\(\widehat{x}=\dfrac{\overrightarrow{x}}{\left| x\right| }\).

Where \({\left| x\right| }\) denotes the magnitude of vector x.

The measure of unit vectors is one(1). We should be careful that if two vectors are said to be unit vectors, then they don’t need to be equal. They might have an equal magnitude but can vary in their direction.

Learn about Sets and Complex Numbers

3. What is a Position Vector?

A position vector is a vector that symbolizes either the position or the location of any given point with respect to any arbitrary reference point like the origin. The direction of the position vector always points from the origin of that vector toward a given point.

Position vectors help us to determine the position along with the direction of movement of the vectors in a 3D dimension Cartesian system.

Consider O as the reference/ origin point and Y be an arbitrary point in space then the vector \(\vec{OY}\) is known as the position vector of the point Y.

4. What is an Equal Vector?

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

An equal vector is defined when two vectors or more than two vectors possess the same magnitude, as well as the same direction.

In the above diagram, the vectors \(\vec{XY}\) and vector \(\vec{MN}\) are equal as they both have the same magnitude as well as direction.

5. What is a Negative Vector?

A negative vector can be defined when one vector is supposed to be the negative of another vector if they have equal magnitudes with opposite directions. Consider there are two vectors P and Q, such that these vectors have the same magnitude but are opposite in direction then these vectors can be presented by:

P = – Q

Also, P and Q are said to be the negative vector to one another.

6. What is a Collinear Vector?

A collinear vector can be defined when two or more than two vectors are parallel to one another irrespective of the magnitude or the direction. The parallel nature of vectors indicates that they never meet or intersect with each other. Consider the below image to understand the same.

Thus, we can estimate any two vectors as collinear vectors if and only if these two vectors are either along the identical line or the vectors are parallel to one another in the same direction/opposite direction. Therefore collinear vectors are also known as parallel vectors.

7. What is a Co-initial Vector?

Co-initial vectors come under the type of vectors wherein two or more than two separate vectors have alike/same initial points. This states that in this type of vector, all vectors begin from the same initial position i.e. the origin spot is identical for the vectors.

For example, if we consider two vectors namely \(\vec{AX}\) and \(\vec{AY}\) as shown below then these vectors are termed co-initial vectors as they both possess a similar initial point that is A.

8. What are Like and Unlike Vectors?

Like vectors are vectors that have the same direction. On the other hand, if the vectors possess opposite directions w.r.t to one another then they are said to be unlike vectors.

9. What is a Co-Planar Vector?

Co-planar vectors are the type of vectors where three or more than three vectors rest in the same plane or lie in the parallel plane.

10. What is a Displacement Vector?

A displacement vector can be defined if a given point is displaced from position say P to Q then the displacement PQ draws a vector \(\vec{PQ}\).

11. What is an Orthogonal Vector?

Two or more than two vectors in space are considered to be orthogonal if the angle between them is 90 degrees.

12. What are Concurrent vectors?

Concurrent vectors are those types of vectors that pass through the same point. This implies that a concurrent vector arrangement is a set of two or more vectors whose lines of action meet at a point.

Learn about Scalar Triple Product

Vectors can be used in many ways, and there are some basic operations you can do with them. These include adding, subtracting, and multiplying vectors. Some operations can be done just by looking at the direction and length of the vectors (geometrically), without using coordinates.

Here are the main vector operations:

➕ 1. Addition of Vectors

To add two vectors, you place them head to tail and draw a new vector from the start of the first to the end of the second.
This new vector is the result of the addition.

➖ 2. Subtraction of Vectors

To subtract vectors, you reverse the direction of the vector you’re subtracting and then add it to the other vector using the head-to-tail method.

✖️ 3. Scalar Multiplication

This means multiplying a vector by a number (called a scalar).

• It changes the length of the vector.
• The direction stays the same (or flips if the number is negative).

 4. Multiplication of Vectors

There are two main ways to multiply vectors:

• Dot Product (Scalar Product):
   
  • Gives a number (not a vector).
  • Shows how much two vectors point in the same direction.
• Cross Product (Vector Product):
  • Gives a new vector.
  • The new vector is perpendicular to both original vectors.

5. Scalar Triple Product

This involves three vectors.
You first take the cross product of two vectors, and then take the dot product of the result with the third vector.

• It gives a single number (scalar).
• It is useful for finding the volume of a 3D shape formed by the three vectors.

Applications of Vectors

Vectors are used in many areas of science, engineering, and daily life. They help us understand things that have both direction and size, like force, speed, and motion.

Here are some easy-to-understand examples of how vectors are useful:

 1. Moving Objects with Force

Vectors show the direction and strength of force applied to move something. This helps us know which way and how hard to push or pull.

 2. Gravity on Moving Objects

When something moves up or down (like throwing a ball), vectors help explain how gravity affects it.

 3. Motion in a Plane

If an object moves in a flat area (like a car turning on a road), vectors can describe its direction and speed.

 4. Force in 3D Space

Vectors help show how a force is applied in all three directions (length, width, and height) at the same time.

 5. Engineering and Structures

Engineers use vectors to check if buildings, bridges, or machines can handle the forces acting on them — whether they will stand or break.

 6. Use in Oscillators

Vectors help in analyzing devices that move back and forth, like a pendulum or electrical circuits.

 7. Quantum Mechanics

In advanced physics, vectors help describe the behavior of tiny particles that we can’t see with our eyes.

 8. Flow of Liquids

In fluid mechanics, vectors show the speed and direction of how liquid moves through a pipe or a river.

 9. General Relativity

In Einstein's theory of general relativity, vectors are used to explain things like gravity and the shape of space-time.

 10. Sound and Wave Propagation

Vectors are used to understand how sound waves or electric signals move from one place to another.

Properties of Types of Vectors

We can apply different mathematical operations to vectors such as addition, subtraction, and multiplication. The different properties of vectors are listed below:

• Addition of vectors is commutative and associative
• Dot Product of two vectors is a scalar and lies in the plane of the two vectors.
• Cross product of two vectors is a vector, which is perpendicular to the plane containing these two vectors.
• \(\vec{A}\).\(\vec{B}\) = \(\vec{B}\).\(\vec{A}\)
• \(\vec{A}\) × \(\vec{B}\) ≠ \(\vec{B}\) × \(\vec{A}\)
• \(\widehat{i}\).\(\widehat{i}\) = \(\widehat{j}\).\(\widehat{j}\) = \(\widehat{k}\).\(\widehat{k}\) = 1
• \(\widehat{i}\).\(\widehat{j}\) = \(\widehat{j}\).\(\widehat{k}\) = \(\widehat{k}\).\(\widehat{i}\) = 0
• \(\widehat{i}\) × \(\widehat{i}\) = \(\widehat{j}\) × \(\widehat{j}\) = \(\widehat{k}\) × \(\widehat{k}\) = 0
• \(\widehat{i}\) × \(\widehat{j}\) = \(\widehat{k}\); \(\widehat{j}\) × \(\widehat{k}\) = \(\widehat{i}\); \(\widehat{k}\) × \(\widehat{i}\) = \(\widehat{j}\)
• \(\widehat{j}\) × \(\widehat{i}\) = -\(\widehat{k}\); \(\widehat{k}\) × \(\widehat{j}\) = -\(\widehat{i}\); \(\widehat{i}\) × \(\widehat{k}\) = -\(\widehat{j}\)

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