Types of Vectors

Vectors are geometrical entities that have magnitude and direction. A vector can be represented by a line with an arrow pointing towards its direction, and its length represents the magnitude of the vector. Having said that vectors are represented by arrows, they have initial points and terminal points. The concept of vectors evolved over a period of 200 years. Vectors are used to represent physical quantities such as displacement, velocity, acceleration, etc..

Further, vectors have numerous applications in physics and engineering. The use of vectors started in the late 19th century with the advent of the field of electromagnetic induction. Here, we will study the definition of vectors along with types of vectors, properties of vectors, along with solved examples for a better understanding.

A vector is a Latin word that means carrier. Vectors are defied from point A to point B. The length of the line between the two points A and B is called the magnitude of the vector, and the direction of the displacement of point A to point B is called the direction of the vector AB. Vectors play an important role in physics. For example, velocity, displacement, acceleration, force are all vector quantities that have a magnitude as well as direction. Vectors are also called Euclidean vectors or Spatial vectors. Vectors have many applications in maths, physics, engineering, and various other fields.

Notation of a vector: The standard form of representation of a vector is: \[\vec A = a\hat i + b \hat j +c \hat k \] where a, b, c are numeric values and \( \hat i, \hat j, \hat k\) are the unit vectors along the x-axis, y-axis, and z-axis respectively.

The vectors are named differently as types of vectors based on their properties such as magnitude, direction, and their relationship with other vectors. These different types of vectors are helpful in performing numerous arithmetic operations and calculations of vectors.

Let us explore a few types of vectors:

• Zero Vectors: Vectors that have 0 magnitude are called zero vectors, denoted by \(\overrightarrow{0}\) = (0,0,0). The zero vector has zero magnitudes and no direction. It is also called the additive identity of vectors.

• Unit Vectors: Vectors that have magnitude equals to 1 are called unit vectors, denoted by \(\hat{a}\). It is also called the multiplicative identity of vectors. The length of unit vectors is 1. It is generally used to denote the direction of a vector.

• Position Vectors: Position vectors are used to determine the position and direction of movement of the vectors in a three-dimensional space. The magnitude and direction of position vectors can be changed relative to other bodies. It is also called the location vector.

• Equal Vectors: Two or more vectors are said to be equal if their corresponding components are equal. Equal vectors have the same magnitude as well as direction. They may have different initial and terminal points but the length and direction must be equal.

• Negative Vectors: A vector is said to be the negative of another vector if they have the same magnitudes but opposite directions. If vectors A and B have equal lengths but opposite directions, then vector A is said to be the negative of vector B or vice versa.

• Parallel Vectors: Two or more vectors are said to be parallel vectors if they have the same direction but not necessarily the same magnitude. The angle between two parallel vectors is zero degrees. The vectors whose angle of direction differs by 180 degrees are called antiparallel vectors, that is, antiparallel vectors have opposite directions.

• Orthogonal Vectors: Two or more vectors in space are said to be orthogonal if the angle between them is 90 degrees. In other words, the dot product of orthogonal vectors is always 0.

• Co-initial Vectors: Vectors that have the same initial point are called co-initial vectors.

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

• The addition of vectors is commutative and associative.
• \(\vec A . \vec B = \vec B. \vec A \)
• \( \vec A \times \vec B\neq \vec B \times \vec A \)
• \(\hat i .\hat i =\hat j.\hat j = \hat k.\hat k = 1 \)
• \(\hat i .\hat j =\hat j.\hat k = \hat k.\hat i = 0 \)
• \(\hat i \times \hat i =\hat j\times \hat j = \hat k\times \hat k = 0 \)
• \(\hat i \times \hat j = \hat k~;~ \hat j\times \hat k = \hat i~;~ \hat k\times \hat i = \hat j \)
• \(\hat j \times \hat i = -\hat k~;~ \hat k \times \hat j = -\hat i~;~ \hat i \times \hat k = -\hat j \)
• The Dot Product of two vectors is a scalar and lies in the plane of the two vectors.
• The cross product of two vectors is a vector, which is perpendicular to the plane containing these two vectors.

Some of the important applications of vectors in real life are listed below:

• The direction in which the force is applied to move the object can be found using vectors.
• To understand how gravity applies as a force on a vertically moving body.
• The motion of a body which is confined to a plane can be obtained using vectors.
• Vectors help in defining the force applied on a body simultaneously in the three dimensions.
• Vectors are used in the field of Engineering, to check if the force is much stronger than the structure and if it will sustain, or collapse.
• In various oscillators, vectors are used.
• Vectors also have their applications in ‘Quantum Mechanics’.
• The velocity of liquid flow in a pipe can be determined in terms of the vector field - for example, fluid mechanics.
• We may also observe them everywhere in general relativity.
• Vectors are used in various wave propagations such as vibration propagation, sound propagation, AC

Related Topics on Types of Vectors:

• Vectors
• Vector Quantities
• Vector Formulas
• Vector Addition as Net Effect
• Resolving a Vector into Components
• Vector Subtraction Calculator
• Vector Projection Formula

Important Notes on Vectors:

• The dot product of orthogonal vectors is always zero.
• The Cross product of parallel vectors is always zero.
• Two or more vectors are collinear if their cross product is zero.
• The magnitude of a vector is a real non-negative value that represents its magnitude.

What Are Vectors in Math?

Vectors are geometrical or physical quantities that possess both magnitude and direction in which the object is moving. The magnitude of the vector indicates the length of the vector. It is generally represented by an arrow pointing in the direction of the vector. The standard form of representation of a vector is: \[\vec A = a\hat i + b \hat j +c \hat k \] where a, b, c are numeric values and \( \hat i, \hat j, \hat k\) are the unit vectors along the x-axis, y-axis, and z-axis respectively.

What Are the Types of Vectors?

The types of vectors are:

• Zero Vectors
• Unit Vectors
• Position Vectors
• Equal Vectors
• Negative Vectors
• Parallel Vectors
• Orthogonal Vectors
• Co-initial Vectors

What is a Negative Vector?

A vector is said to be the negative of another vector if they have the same magnitudes but opposite directions. The angle between a vector and its negative vector differ by 180°. If vectors A and B have equal lengths but opposite directions, then vector A is said to be the negative of vector B or vice versa.

What Is Zero Vector Give Example?

Vectors that have 0 magnitude are called zero vectors, and is denoted by \(\overrightarrow{0}\) = (0,0,0). The zero vector has zero magnitudes and no direction. It is also called the additive identity of vectors.

What Are the Properties of Vectors?

There are several properties of vectors, few of them are:

• The addition of vectors is commutative and associative.
• The additive identity of vectors is the zero vector.
• The additive inverse of a vector is the negative of the vector.
• The scalar multiplication of vectors is associative.

What Are Collinear Vectors?

Collinear vectors are vectors that are parallel/antiparallel to the same line irrespective of their magnitude and direction. The cross-product of collinear vectors is always zero and corresponding components have an equal ratio.

How are Vectors Linearly Independent?

Vectors are said to be linearly independent if there exists a non-trivial linear combination of vectors that is equal to zero. If no such linear combination exists, the vectors are said to be linearly dependent. In other words, a set of vectors {v1, v2, v3, ..., vn} are linearly independent if there exists nontrivial scalars k1, k2, k3, ..., know, such that k1v1 + k2v2 + k3v3 + ... + knvn = 0

When Are the Two Vectors Said to be Parallel Vectors?

Two or more vectors are parallel if they are moving in the same direction. The angle between two parallel vectors is either 0°, or 180°. Also,the cross-product of parallel vectors is always zero.