Types of Vectors | Zero Vector, Unit Vector, Position Vector, Co-initial Vector, Like & Unlike Vectors, Co-planar Vectors, Collinear Vectors, Equal Vectors, Displacement Vector, Negative of a Vector
Vector is a physical quantity that has both direction and magnitude. In other words, the vectors are defined as an object comprising both magnitude and direction. It describes the movement of the object from one point to another. The below figure shows the vector with head, tail, magnitude and direction.
There are 10 different types of vectors that are generally used in maths and science. The various vector types that are covered here are as follows.
Table of Contents
Types of Vectors List
There are 10 types of vectors in mathematics which are:
Zero Vector
Unit Vector
Position Vector
Co-initial Vector
Like and Unlike Vectors
Co-planar Vector
Collinear Vector
Equal Vector
Displacement Vector
Negative of a Vector
All these vectors are extremely important and the concepts are frequently required in mathematics and other higher-level science topics. The detailed explanations on each of these 10 vector types are given below.
Zero Vector
A zero vector is a vector when the magnitude of the vector is zero and the starting point of the vector coincides with the terminal point.
\(\begin{array}{l}\text{In other words, for a vector }\overrightarrow{AB} \text{ the coordinates of the point A are the same as that of the} \\ \text{point B then the vector is said to be a zero vector and is denoted by 0.}\end{array} \)
This follows that the magnitude of the zero vector is zero and the direction of such a vector is indeterminate.
Unit Vector
A vector which has a magnitude of unit length is called a unit vector.
\(\begin{array}{l}\text{Suppose if }\overrightarrow{x} \text{ is a vector having a magnitude x then the unit vector is denoted by } x̂ \\ \text{ in the direction of the vector } \overrightarrow{x} \text{ and has the magnitude equal to 1.}\end{array} \)
It must be carefully noted that any two unit vectors must not be considered as equal, because they might have the same magnitude, but the direction in which the vectors are taken might be different.
Position Vector
\(\begin{array}{l}\text{If O is taken as reference origin and P is an arbitrary point in space then the vector }\overrightarrow{OP} \\ \text{ is called as the position vector of the point.}\end{array} \)
Position vector simply denotes the position or location of a point in the three-dimensional Cartesian system with respect to a reference origin.
Co-initial Vectors
The vectors which have the same starting point are called co-initial vectors.
\(\begin{array}{l}\text{The vectors } \overrightarrow{AB} \text{ and } \overrightarrow{AC} \text{ are called co-initial vectors as they have the same starting point.}\end{array} \)
Like and Unlike Vectors
The vectors having the same direction are known as like vectors. On the contrary, the vectors having the opposite direction with respect to each other are termed to be unlike vectors.
Co-planar Vectors
Three or more vectors lying in the same plane or parallel to the same plane are known as co-planar vectors.
Collinear Vectors
Vectors that lie along the same line or parallel lines are known to be collinear vectors. They are also known as parallel vectors.
Two vectors are collinear if they are parallel to the same line irrespective of their magnitudes and direction. Thus, we can consider any two vectors as collinear vectors if and only if these two vectors are either along the same line or these vectors are parallel to each other in the same direction or opposite direction. For any two vectors to be parallel to one another, the condition is that one of the vectors should be a scalar multiple of another vector. The below figure shows the collinear vectors in the opposite direction.
Equal Vectors
Two or more vectors are said to be equal when their magnitude is equal and also their direction is the same.
The two vectors shown above, are equal vectors as they have both direction and magnitude equal.
Displacement Vector
\(\begin{array}{l}\text{If a point is displaced from position A to B then the displacement AB represents a vector } \overrightarrow{AB} \\ \text{which is known as the displacement vector.}\end{array} \)
Negative of a Vector
If two vectors are the same in magnitude but exactly opposite in direction then both the vectors are negative of each other. Assume there are two vectors a and b, such that these vectors are exactly the same in magnitude but opposite in direction then these vectors can be given by
a = – b
Neeraj Anand, Param Anand
Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations.
In parallel, he started a Technical Publication "ANAND TECHNICAL PUBLISHERS" in 2002 and Educational Newspaper "NATIONAL EDUCATION NEWS" in 2014 at Jalandhar. Now he is a Director of leading publication "ANAND TECHNICAL PUBLISHERS", "ANAND CLASSES" and "NATIONAL EDUCATION NEWS".
He has published more than hundred books in the field of Physics, Mathematics, Computers and Information Technology. Besides this he has written many books to help students prepare for IIT-JEE and AIPMT entrance exams. He is an executive member of the IEEE (Institute of Electrical & Electronics Engineers. USA) and honorary member of many Indian scientific societies such as Institution of Electronics & Telecommunication Engineers, Aeronautical Society of India, Bioinformatics Institute of India, Institution of Engineers. He has got award from American Biographical Institute Board of International Research in the year 2005.
CBSE Class 12 Maths Syllabus 2025-26 with Marks Distribution
The table below shows the marks weightage along with the number of periods required for teaching. The Maths theory paper is of 80 marks, and the internal assessment is of 20 marks which totally comes out to be 100 marks.
CBSE Class 12 Maths Syllabus And Marks Distribution 2023-24
Max Marks: 80
No.
Units
Marks
I.
Relations and Functions
08
II.
Algebra
10
III.
Calculus
35
IV.
Vectors and Three – Dimensional Geometry
14
V.
Linear Programming
05
VI.
Probability
08
Total Theory
80
Internal Assessment
20
Grand Total
100
Unit-I: Relations and Functions
1. Relations and Functions
Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions.
2. Inverse Trigonometric Functions
Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions.
Unit-II: Algebra
1. Matrices
Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operations on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).
2. Determinants
Determinant of a square matrix (up to 3 x 3 matrices), minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.
Unit-III: Calculus
1. Continuity and Differentiability
Continuity and differentiability, derivative of composite functions, chain rule, derivative of inverse trigonometric functions like sin-1 x, cos-1 x and tan-1 x, derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives.
2. Applications of Derivatives
Applications of derivatives: rate of change of quantities, increasing/decreasing functions, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).
3. Integrals
Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.
Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.
4. Applications of the Integrals
Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only)
5. Differential Equations
Definition, order and degree, general and particular solutions of a differential equation. Solution of differential equations by method of separation of variables, solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:
dy/dx + py = q, where p and q are functions of x or constants.
dx/dy + px = q, where p and q are functions of y or constants.
Unit-IV: Vectors and Three-Dimensional Geometry
1. Vectors
Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.
2. Three – dimensional Geometry
Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, skew lines, shortest distance between two lines. Angle between two lines.
Unit-V: Linear Programming
1. Linear Programming
Introduction, related terminology such as constraints, objective function, optimization, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).
Unit-VI: Probability
1. Probability
Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean of random variable.
Students can go through the CBSE Class 12 Syllabus to get the detailed syllabus of all subjects. Get access to interactive lessons and videos related to Maths and Science with ANAND CLASSES’S App/ Tablet.
Frequently Asked Questions on CBSE Class 12 Maths Syllabus 2025-26
Q1
Is Calculus an important chapter in the CBSE Class 12 Maths Syllabus?
Yes, Calculus is an important chapter in the CBSE Class 12 Maths Syllabus. It is for 35 marks which means that if a student is thorough with this chapter will be able to pass the final exam.
Q2
How many units are discussed in the CBSE Class 12 Maths Syllabus?
In the CBSE Class 12 Maths Syllabus, about 6 units are discussed, which contains a total of 13 chapters.
Q3
How many marks are allotted for internals in the CBSE Class 12 Maths syllabus?
About 20 marks are allotted for internals in the CBSE Class 12 Maths Syllabus. Students can score it with ease through constant practice.
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