Scalar or Dot Product Of Two Vectors | Projection of Vector | Definition, Properties, Formulas and Examples | Class 12 Math Notes Study Material Download Free PDF
Vector is a quantity that has both magnitude and direction. Some mathematical operations can be performed on vectors such as addition and multiplication. The multiplication of vectors can be done in two ways, i.e. dot product and cross product. In this article, you will learn the dot product of two vectors with the help of examples.
The definition of dot product can be given in two ways, i.e. algebraically and geometrically. Algebraically, the dot product is defined as the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the two vectors’ Euclidean magnitudes and the cosine of the angle between them. Both the definitions are equivalent when working with Cartesian coordinates. However, the dot product of two vectors is the product of the magnitude of the two vectors and the cos of the angle between them. To recall, vectors are multiplied using two methods
scalar product of vectors or dot product
vector product of vectors or cross product
The difference between both the methods is just that, using the first method, we get a scalar value as resultant and using the second technique the value obtained is again a vector in nature.
Table of Contents
Dot Product of Vectors
The scalar product of two vectors a and b of magnitude |a| and |b| is given as |a||b| cos θ, where θ represents the angle between the vectors a and b taken in the direction of the vectors. We can express the scalar product as:
a.b=|a||b|cosθ
where |a| and |b| represent the magnitude of the vectors a and b while cos θ denotes the cosine of the angle between both the vectors and a.b indicate the dot product of the two vectors.
In the case, where any of the vectors is zero, the angle θ is not defined and in such a scenario a.b is given as zero.
Projection of Vectors
BP is known to be the projection of a vector a on vector b in the direction of vector b given by |a| cos θ.
Similarly, the projection of vector b on a vector a in the direction of the vector a is given by |b| cos θ.
Projection of vector a in direction of vector b is expressed as
\(\begin{array}{l} BP = \frac{a.b}{|b|}\end{array} \)
Thus, we see that the dot product of two vectors is the product of magnitude of one vector with the resolved component of the other in the direction of the first vector.
Dot Product Properties of Vector
Property 1: Dot product of two vectors is commutative i.e. a.b = b.a = ab cos θ.
Property 2: If a.b = 0 then it can be clearly seen that either b or a is zero or cos θ = 0. \(\begin{array}{l}\Rightarrow \theta=\frac{\pi}{2}.\end{array} \) It suggests that either of the vectors is zero or they are perpendicular to each other.
Property 3: Also we know that using scalar product of vectors (pa).(qb)=(pb).(qa)=pq a.b
Property 4: The dot product of a vector to itself is the magnitude squared of the vector i.e. a.a = a.a cos 0 = a2
Property 5: The dot product follows the distributive law also i.e. a.(b + c) = a.b + a.c
Property 6: In terms of orthogonal coordinates for mutually perpendicular vectors it is seen that \(\begin{array}{l}\hat{i}.\hat{i} = \hat{j}.\hat{j}= \hat{k}.\hat{k} = 1\end{array} \)
Property 7: In terms of unit vectors, if \(\begin{array}{l}a = a_{1}\hat{i}+ a_{2}\hat{j}+a_{3}\hat{k} \ and\ b = b_{1}\hat{i}+ b_{2}\hat{j}+b_{3}\hat{k}\end{array} \) then \(\begin{array}{l}a.b = (a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}).(b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k})\end{array} \)
Example 1: Let there be two vectors [6, 2, -1] and [5, -8, 2]. Find the dot product of the vectors.
Solution:
Given vectors: [6, 2, -1] and [5, -8, 2] be a and b respectively.
a.b = (6)(5) + (2)(-8) + (-1)(2)
a.b = 30 – 16 – 2
a.b = 12
Example 2: Let there be two vectors |a|=4 and |b|=2 and θ = 60°. Find their dot product.
Solution:
a.b = |a||b|cos θ
a.b = 4.2 cos 60°
a.b = 4.2 × (1/2)
a.b = 4
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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