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Dot Product Formula
Dot Product Formula is an important concept of Physics and Mathematics.
Physics is a branch of science that studies the structure of matter and the interactions of the universe’s fundamental constituents. Physics (from the Greek term ‘physikos’) includes all aspects of nature, both macroscopic and submicroscopic. Its research interests span the nature and origin of gravitational, electromagnetic, and nuclear force fields in addition to the behaviour of objects under various forces. Its ultimate goal is to develop a few comprehensive principles that bring all of these disparate phenomena together and explain them.
Physics is a fundamental Physical Science. Until recently, the terms Physics and Natural Philosophy were used interchangeably to refer to the knowledge of Science whose goal is to discover and formulate the fundamental laws of nature. As the modern sciences evolved and became more specialised, the term “Physics” came to refer to the portion of physical science that was not covered by astronomy, chemistry, geology, or engineering. However, Physics is important in all natural sciences, and all of them have branches in which physical laws and measurements are given special attention, with names like Astrophysics, Geophysics, Biophysics, and even Psychophysics. Physics is fundamentally the Science of matter, motion, and energy. Its laws are typically expressed in Mathematics with economy and precision.
Both experiment, or the observation of phenomena under precisely controlled conditions, and theory, or the formulation of a unified conceptual framework, play critical and complementary roles in the advancement of Physics. Physical experiments produce measurements, which are compared to the predicted outcome by theory. A physics law is a theory that reliably predicts the outcomes of experiments to which it is applicable. A law, however, is always subject to modification, replacement, or restriction to a more limited domain if a subsequent experiment requires it.
The ultimate goal of Physics is to discover a unified set of laws governing matter, motion, and energy at small (microscopic) subatomic distances, the human (macroscopic) scale of everyday life, and beyond (e.g., those on the extragalactic scale). This lofty goal has been met to a large extent. Although a completely unified theory of physical phenomena has not yet been achieved (and may never be), a surprisingly small set of fundamental physical laws appears to be capable of accounting for all known phenomena. Classical Physics, which existed until around the turn of the twentieth century, can account for the motions of macroscopic objects that move slowly in comparison to the speed of light. These laws are modified by modern developments in relativity and quantum mechanics insofar as they apply to higher speeds, very massive objects, and the tiniest elementary constituents of matter, such as electrons, protons, and neutrons.
Due to the fact that Physics elucidates the most basic fundamental questions in nature on which there can be agreement, it is not surprising that it has had a profound impact on other fields of Science, Philosophy, the developed world’s worldview, and, of course, technology.
Indeed, whenever a branch of Physics has matured to the point where its basic elements can be understood in general principles, it has progressed from basic to applied physics, and then to technology. Thus, Applied Physics constitutes almost all current activity in classical physics, and its contents are at the heart of many branches of engineering. Modern physics discoveries are increasingly being translated into technical innovations and analytical tools for related disciplines. Nuclear and Biomedical Engineering, Quantum Chemistry and Quantum Optics, radio, Xray, and gammaray astronomy, and analytic tools like radioisotopes, spectroscopy, and lasers are all emerging fields that stem directly from basic physics.
Aside from its specific applications, Physics—particularly Newtonian mechanics—has become the prototype of the scientific method, with its experimental and analytic methods occasionally (and sometimes inappropriately) imitated in fields unrelated to the Physical Sciences. Some organisational aspects of Physics have been imitated in largescale scientific projects, such as Astronomy and Space Research, based in part on the successes of World War II radar and atomicbomb projects.
The earlier designation of Physics as Natural Philosophy attests to its significant influence on branches of Philosophy concerned with the conceptual basis of human perceptions and understanding of nature, such as epistemology. Modern Philosophy of Science is concerned primarily, but not exclusively, with the foundations of Physics. Determinism, the philosophical doctrine that the universe is a vast machine that operates with strict causality and whose future is determined in every detail by its current state, is based on Newtonian mechanics, which adheres to that principle. Furthermore, the schools of Materialism, Naturalism, and Empiricism all saw Physics as a model for philosophical inquiry. The logical positivists take an extreme stance, demanding that all significant statements be written in the language of Physics due to their radical distrust of the reality of anything not directly observable.
What is Dot Product?
Dot product Formula is one method of multiplying two or more vectors. The Dot Product Formula of vectors produces a scalar quantity. As a result, the dot product is also referred to as a scalar product. It is the sum of the products of the corresponding entries of two number sequences. Geometrically, it is the product of two vectors’ Euclidean magnitudes and the cosine of the angle between them. Vector Dot Product Formula has numerous applications in Geometry, Mechanics, Engineering, and Astronomy.
Dot Product Definition
The Dot Product Formula definition is very important for students to understand. The vector dot product equals the product of the magnitudes of the two vectors and the cosine of the angle between the two vectors. The Dot Product Formula of two vectors lies in the same plane as the two vectors. The Dot Product Formula can be a positive or negative real number, or it can be zero.
Dot Product Formula for Vectors
The Dot Product Formula for Vectors is also very important for Class 12 Mathematics. Mathematics is an important subject for Class 12 students. A basic understanding of Mathematics at the school level is critical for students’ development of reasoning and logical thinking abilities. Students can learn to solve realworld problems by understanding mathematical concepts. Mathematics is used in almost every field in some way. The knowledge of Mathematics can be useful for pursuing careers in a variety of fields. With the help of its principles, students can pursue careers in Astronomy, Astrophysics, Statistics, Weather Forecasting, and other fields. Mathematics is taught in schools beginning in primary school and is one of the core subjects in a curriculum. Mathematics fundamentals are important for a student’s overall development.
The examinations for Class 12 students are administered by the Central Board of Secondary Education (CBSE). Without a doubt, Class 12 is one of the most important milestones in a student’s life. Class 12 grades can be a deciding factor in choosing a career. Learning effectively in Class 12 is advantageous for preparing for various competitive exams such as JEE and NEET.
Class 12 board exams are critical for students to take the first step toward their future studies. Students in Class 12 should prepare thoroughly for the upcoming board exams. It is necessary to create a study plan in order to master Class 12 subjects. Students in Class 12 should concentrate on learning each subject thoroughly. Students in Class 12 of the Central Board of Secondary Education (CBSE) must practice mathematical topics on a regular basis. It is critical to solve exercise questions on a regular basis. Regularly answering exercise questions can help students prepare for the Mathematics exam.
Students are advised to study the topics of Class 12 Mathematics from the standpoint of the board exam. Dot Product Formula should also be studied in this manner. All of the topics are essential for students to prepare for. Class 12 students must use the most recent CBSE Mathematics syllabus to get an overview of all the topics and subtopics covered in Mathematics.
The topics covered in Class 12 are extremely broad and necessitate frequent revision and practice. Solving exercise problems can be difficult for students at times. It is advised that learners use NCERT solutions to solve exercise questions.
Studying Dot Product Formula well will help students to score well in both Physics and Maths Class 12 exams. The Dot Product Formula can be easily understood with the help of Extramarks’ resources.
Geometrical Meaning of Dot Product
The Dot Product Formula of two vectors is calculated by multiplying the component of one vector in the direction of the other by the magnitude of the other vector. To comprehend the vector Dot Product Formula, one must first understand how to calculate the magnitude of two vectors and the angle between two vectors in order to calculate the projection of one vector over another vector.
Magnitude of A Vector
A vector has a magnitude and a direction. The magnitude of a vector is equal to the square root of the sum of the squares of the vector’s individual constituents. A vector’s magnitude is a positive number. For a vector,
 a  = a 1 I + a 2 j + a 3 k, the magnitude is  a  and is given by the formula,  a  = a 2 1 + a 2 2 + a 2 3
Projection of a Vector
The Dot Product Formula can be used to find the component of one vector pointing in the direction of another. The length of the shadow of one vector over another vector is the vector projection of one vector over another vector. It is calculated by multiplying the magnitude of the given vectors by the cosine of the angle between them. A scalar value is the result of a vector projection formula.
Angle Between Two Vectors Using Dot Product
The cosine of the angle between two vectors is used to calculate the angle between two vectors. The cosine of the angle between two vectors is equal to the sum of the products of the two vectors’ individual constituents divided by the product of their magnitudes.
Working Rule to Find The Dot Product of Two Vectors
If the two vectors are written in terms of unit vectors, i, j, k, along the x, y, z axes, then the scalar product is obtained as follows:
If →a=a1ˆi+a2ˆj+a3ˆk
and →b=b1ˆi+b2ˆj+b3ˆk
, then
→a.→b
= (a1ˆi+a2ˆj+a3ˆk)(b1ˆi+b2ˆj+b3ˆk)
= (a1b1)(ˆi.ˆi)+(a1b2)(ˆi.ˆj)+(a1b3)(ˆi.ˆk)+(a2b1)(ˆj.ˆi)+(a2b2)(ˆj.ˆj)+(a2b3(ˆj.ˆk)+(a3b1)(ˆk.ˆi)+(a3b2)(ˆk.ˆj)+(a3b3)(ˆk.ˆk)
= a1b1
+ a2b2
+ a3b3
Matrix Representation of Dot Product
When vectors are represented as row or column matrices, the Dot Product Formula is simple to compute. The first vector’s transpose matrix is obtained as a row matrix. The matrix is multiplied. The row matrix and column matrix are multiplied to obtain the sum of the product of the two vectors’ corresponding components.
Properties of Dot Product
The following are the properties of the Dot Product Formula of vectors.
 Commutative property of Dot Product Formula
 Distributive property of Dot Product Formula
 Natural property of Dot Product Formula
 General properties of Dot Product Formula
 Vector identities
Commutative property of Dot Product:
The Commutative property of the Dot Product Formula is given below. After learning the commutative property students must solve questions related to this concept to understand it thoroughly. :
Commutative Property
a .b = b.a
a.b =a bcos θ
a.b =bacos θ
Distributivity of Dot Product
The Distributivity of the Dot Product Formula is given below:
a.(b + c) = a.b + a.c
To understand this property fully, students must solve practice questions and examples on this property. They can further refer to the resources provided by Extramarks to gain more knowledge on this topic.
Nature of Dot Product
While it may be easy to overlook, the Dot Product Formula is defined between two vectors of the same dimension in both definitions (this is most obvious in the component description of the Dot Product Formula, which requires us to pair each component in u with one in v). When one computes the Dot Product Formula in both definitions, the result is a scalar.
For vectors of the same dimension, the Dot Product Formula is defined. When it has been defined,
vector∙vector=scalar.
Other Properties of Dot Product
Various other properties of the Dot Product Formula are given below for students’ understanding. All the properties are very important from an exam point of view.
 Bilinear Property
a.(rb + c) = r.(a.b) + (a.c)
 Scalar Multiplication Property
(xa) . (yb) = xy (a.b)
 NonAssociative Property
Since the Dot Product Formula between a scalar and a vector is not allowed.
 Orthogonal Property
Two vectors are orthogonal only when a.b = 0
Vector Identities
Some of the Vector Identities are given below:
A · (B × C) = C · (A × B) = B · (C × A)
A × (B × C) = B(A · C) − C(A · B)
(A × B) × C = B(A · C) − A(B · C)
∇×∇ f = 0
∇ · (∇ × A) = 0
∇ · ( f A) = (∇ f ) · A + f (∇ · A)
∇ × ( f A) = (∇ f ) × A + f (∇ × A)
∇ · (A × B) = B · (∇ × A) − A · (∇ × B)
∇(A · B) = (B · ∇)A + (A · ∇)B + B × (∇ × A) + A × (∇ × B)
∇ · (A B) = (A · ∇)B + B(∇ · A)
∇ × (A × B) = (B · ∇)A − (A · ∇)B − B(∇ · A) + A(∇ · B)
∇ × (∇ × A) = ∇(∇ · A) − ∇2 A
Dot Product of Unit Vectors
By taking the unit vectors I the Dot Product Formula of the unit vectors is investigated.
Along the xaxis, j along the yaxis, and k along the zaxis, in that order. The Dot Product Formula of unit vectors I j, k follows the same rules as the Dot Product Formula of vectors. Due to the fact that the angle between the same vectors is equal to 0o, their Dot Product Formula is equal to 1. And the angle formed by two perpendicular vectors is 90o, so their Dot Product Formula is 0.
I = j
I = 0
Applications of Dot Product
The scalar product is used in the calculation of work. The work is defined as the product of the applied force and displacement. When force is applied at an angle to displacement, work done is given as the Dot Product Formula of force and displacement as W = f d cos. The Dot Product Formula can also be used to determine whether two vectors are orthogonal.
The Dot Product Formula, also known as the scalar product, is a method of multiplying two vectors. The Dot Product Formula is the geometric product of the vectors’ lengths and the cosine angle between them.
 a  b  cos
It is a directionless scalar quantity. It is easily calculated by adding the product of the two vectors’ components.
If a is equal to a 1 I + a 2 j + a 3 k and b is equal to b 1 I + b 2 j + b 3 k, then a. b = a 1 b 1 + a 2 b 2 + a 3 b 3
Dot Product of Vectors Examples
Learning becomes easier with the help of examples and practice questions. To understand any topic students must solve a lot of questions. Students can refer to the resources provided by Extramarks if they face any difficulties in solving questions about the Dot Product Formula of Vectors. Extramarks provides various resources for the benefit of students. The study tools provided by Extramarks include Revision notes, NCERT solutions, past years’ papers solutions etc. Students can download all these resources from the website and mobile application of Extramarks. Further, Extramarks also provides all the resources in Hindi medium too. Students can rely on the resources provided as they are written by expert subject teachers.
Practice Questions on Dot Product
Practice questions help students understand the topic well. Students must solve a lot of practice questions on Dot Product Formula so they can answer questions in exams effectively. They can use the various study tools offered by Extramarks to solve the practice questions. With the help of the study tools offered, students will be able to score well in both Physics and Mathematics in Class 12. Dot Product Formula practice questions must be solved by students in this manner to enhance their preparation.
FAQs (Frequently Asked Questions)
1. What is Dot Product Formula?
Dot Product Formula is one method of multiplying two or more vectors. The Dot Product Formula of vectors produces a scalar quantity. As a result, the Dot Product Formula is also referred to as a scalar product. It is the sum of the products of the corresponding entries of two number sequences.
2. What are some of the Vector Identities?
Some of the Vector identities are given below:
A · (B × C) = C · (A × B) = B · (C × A)
A × (B × C) = B(A · C) − C(A · B)
(A × B) × C = B(A · C) − A(B · C)
∇×∇ f = 0
∇ · (∇ × A) = 0
∇ · ( f A) = (∇ f ) · A + f (∇ · A)
∇ × ( f A) = (∇ f ) × A + f (∇ × A)
∇ · (A × B) = B · (∇ × A) − A · (∇ × B)
∇(A · B) = (B · ∇)A + (A · ∇)B + B × (∇ × A) + A × (∇ × B)
∇ · (A B) = (A · ∇)B + B(∇ · A)
∇ × (A × B) = (B · ∇)A − (A · ∇)B − B(∇ · A) + A(∇ · B)
∇ × (∇ × A) = ∇(∇ · A) − ∇2 A
3. What is the Commutative Property of the Dot Product Formula?
The Commutative Property of the Dot Product Formula is given below:
a .b = b.a
a.b =a bcos θ
a.b =bacos θ
4. What is the Distributive Property of the Dot Product Formula?
The Distributive Property of the Dot Product Formula is given below:
a.(b + c) = a.b + a.c
5. From where can students download the resources provided by Extramarks?
Students can download the various resources provided by Extramarks from their website and mobile application.
6. What are the various study tools provided by Extramarks?
Extramarks provides various resources such as revision notes, NCERT solutions and past years’ papers solutions to students.