JEE

# JEE Chapter - Vector Algebra

Get interactive courses taught by top teachers

## Vector Algebra

Algebraic operations on vectors are included in vector algebra. Vector algebra is the branch of mathematics that deals with the magnitude and direction of vectors. In physics and engineering, vector algebra is used to execute addition and multiplication operations on physical values represented as vectors in three-dimensional space.

In this article, we will learn about vector algebra and its operations, types of vectors along with the help of examples and practice questions for better understanding and knowledge.

### Important topics of Vector Algebra

• Vectors

• Types of Vectors

• Vector Algebra

• Section Formula

• Product of two Vectors

### What is Vector Algebra?

Many algebraic operations involving vectors are performed using vector algebra. The word vector comes from the Latin word vector, which means "carrier." Vectors transport information from point A to point B. The magnitude of the vector is the length of the line between two points A and B, and the direction of the vector AB is the direction of displacement of point A to point B. Euclidean vectors and spatial vectors are other names for vectors. Vectors are used in arithmetic, physics, engineering, and a variety of other professions.

Image: Vector Components

In mathematics, a vector is a geometric entity with both magnitude and direction. Vectors have a starting point at which they begin and a terminal point that indicates the point's final position. In vector algebra, several algebraic operations such as addition, subtraction, and multiplication can be done. Many physical quantities, such as velocity, displacement, acceleration, and force, are vector values, meaning they have both a magnitude and a direction.

### Representation of Vectors

Vectors are usually represented in bold lowercase such as a or using an arrow over the letter as $\stackrel{\to }{a}$$\vec{a}$. Vectors can also be denoted by their initial and terminal points with an arrow above them, for example, vector AB can be denoted as $\stackrel{\to }{AB}$$\overrightarrow{AB}$. The standard form of representation of a vectors is $\stackrel{\to }{A}=a\stackrel{^}{i}+b\stackrel{^}{j}+c\stackrel{^}{k}$$\vec{A}=a \hat{i}+b\hat{j}+c\hat{k}$. Here, a,b,c are real numbers and $\stackrel{^}{i},\stackrel{^}{j},\stackrel{^}{k}$$\hat{i}, \hat{j}, \hat{k}$ are the unit vectors along the x-axis, y-axis, and z-axis respectively.

Image: Vector representation

The tail of a vector is also known as the beginning, whereas the head is known as the end. The movement of an object from one location to another is described by vectors. Vectors are represented as ordered pairs in the cartesian coordinate system. An 'n-tuple' can also be used to represent vectors in 'n' dimensions. Vectors are also identified with a tuple of components which are the scalar coefficients for a set of basis vectors and the basis vectors are denoted as e1 = (1,0,0), e2 = (0,1,0), e3 = (0,0,1)

### Magnitude of Vectors

The square root of the sum of the squares of a vector's components can be used to calculate its magnitude. Assume, if (x,y,z) are the components of a vector A, then the magnitude formula of A is given by,

$|z|=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$$\left | z \right |=\sqrt{x^2+y^2+z^2}$

The magnitude of a vector is a scalar value.

### Types of Vectors - Vector Algebra

Different types of vectors are utilised in vector algebra for various algebraic operations. Based on their magnitude, direction, and relationships with other vectors, vectors are classified into different types. Let's look at a few different types of vectors and their properties accordingly:

### Zero Vectors

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

### Unit Vectors

Vector having a magnitude equal to 1 are called the unit vectors, denoted by $\stackrel{^}{a}$$\hat{a}$. It is also called the multiplicative identity of vectors and the magnitude of a unit vector is 1. It is usually used to denote the direction of a vector.

### Position Vectors

In three-dimensional space, position vectors are used to determine the position and direction of movement of vectors. Position vectors' magnitude and direction can be modified in relation to other bodies, and it is also known as the position vector.

### Equal Vectors

If the corresponding components of two or more vectors are identical, they are said to be equal. The magnitude and direction of equal vectors are the same. Although the initial and terminal points may differ, the magnitude and direction must be equal.

### Negative Vector

A vector is said to be the negative of any other vector if they have the same magnitudes but are opposite in direction. Consider if vectors A and B have equal magnitude but opposite directions, then vector A is said to be the negative of vector B or vice-versa.

### Parallel Vectors

If two or more vectors have the same direction but not necessarily the same magnitude, they are said to be parallel vectors. The angles of parallel vectors' directions differ by zero degrees. Antiparallel vectors are those whose direction angles differ by 180 degrees, i.e., antiparallel vectors have opposite directions.

### Orthogonal Vectors

If the angle between two or more vectors in space is 90 degrees, they are said to be orthogonal. In other words, the dot product of orthogonal vectors is always 0. a·b = |a|·|b|cos90° = 0.

### Co-initial Vectors

Co-initial vectors are two vectors that have the same initial point.

### Operations in Vector Algebra

Some basic vector algebra operations can be done geometrically without the use of a coordinate system. Addition, subtraction, and multiplication by a scalar are some of the operations that can be performed on vectors. The dot product and the cross product of vectors are also two other ways of multiplying vectors. The following are the various operations in vector algebra.

• Subtraction of Vectors

• Scalar Multiplication

• Scalar Triple Product of Vectors

• Multiplication of Vectors

Let us understand each of these operations in vector algebra in the below paragraphs.

Let us consider there are two vectors P and Q, then the sum of these two vectors can be performed when the tail of vector Q meets with the head of vector A. And during this addition, the magnitude and direction of the vectors should not change. The vector addition follows two important laws, which are;

Commutative Law: P + Q = Q + P

Associative Law: P + (Q + R) = (P + Q) + R

### Subtraction of Vectors

Here, the direction of other vectors is reversed and then the addition operation is performed on both the vectors given. If P and Q are the vectors for which the subtraction method must be used, we must invert the direction of another vector, such as Q, to make it -Q. The vectors P and -Q must now be added. The vectors' directions are thus opposite, but their magnitude remains the same.

P – Q = P + (-Q)

### Multiplication of Vectors

The scalar multiplication is represented by kA when k is a scalar quantity multiplied by a vector A. If k is positive, the direction of vector kA is the same as the direction of vector A; however, if k is negative, the direction of vector kA is the opposite of the direction of vector A and the magnitude of the vector kA is given by |kA|.

### Dot Product

A scalar product is often referred to as a dot product. Between two vectors, a dot(.) is used to symbolise it. Two equal-length coordinate vectors are multiplied to get a single integer. In other words, when two vectors are scalar products, the outcome is either a number or a scalar quantity. If P and Q are two vectors, then the dot product of both vectors is as follows:

P.Q = |P| |Q| cos θ

If P and Q are both in the same direction, i.e. θ = 0°, then;

P.Q = |P| |Q|

If P and Q are both orthogonal, i.e. θ = 90°, then;

P.Q = 0 $\left[$$[$since cos 90° = 0$\right]$$]$

In vector algebra, if two vectors are given as;

P = [P1,P2,P3,P4,….,Pn] and Q = [Q1,Q2,Q3,Q4,….,Qn]

Then their dot product is given by;

P.Q = P1Q1+P2Q2+P3Q3+……….PnQn

### Cross Product

The multiplication sign(x) between two vectors denotes a cross product. It's a three-dimensional system with a binary vector operation. If P and Q are two independent vectors, the outcome of their cross product (P x Q) is perpendicular to both vectors and normal to the plane in which they are both located. It is represented by;

P x Q = |P| |Q| sin θ$\stackrel{^}{n}$$\hat{n}$

### Applications of Vector Algebra

In the fields of physics and mathematics, vector algebra has several applications. Quantities with both direction and magnitude are dealt with in vector algebra. Many quantities, such as velocity, acceleration, and force, must be represented as mathematical expressions and can be represented as vectors. The following are some examples of vector algebra applications.

• Vectors are one of the most significant factors in the study of partial differential equations and differential geometry.

• Vectors are particularly valuable in the study of many domains in physics and engineering, such as electromagnetic fields, gravitational fields, and fluid flow.

• Vector algebra is handy when finding the component of a force in a certain direction.

• Vector algebra is used in physics to determine the interaction between two or more quantities.

• The dot of one vector is the cross product of the other two vectors in the scalar triple product of vectors. In a scalar triple product, if any two vectors are equal, the scalar triple product is zero. The three vectors a, b, and c are said to be coplanar if the scalar triple product is equal to zero.

Also, a·(b × c) = b·(c × a) = c·(a × b)

### List of Formulae

 Sl.no Name of the Concept Formulae Magnitude of two dimensional Vector |a| = $\sqrt{{a}_{1}^{2}+{a}_{2}^{2}}$$\sqrt{a_{1}^{2} + a_{2}^{2}}$ Magnitude of three dimensional Vector |a| = $\sqrt{{a}_{1}^{2}+{a}_{2}^{2}+{a}_{3}^{2}}$$\sqrt{a_{1}^{2} + a_{2}^{2} + a_{3}^{2}}$ Dot product of the vector $\stackrel{\to }{A}·\stackrel{\to }{B}$$\vec{A} ·\vec{B}$ = |A||B| cosθ 4. Cross product of vector $\stackrel{\to }{A}×\stackrel{\to }{B}$$\vec{A} \times \vec{B}$ = |A||B| sinθ $\stackrel{^}{n}$$\hat{n}$ 5. Cross product of the vector in matrix $\stackrel{\to }{A}×\stackrel{\to }{B}$$\vec{A} \times \vec{B}$ = (b1c2 - c1b2, a1c2 - c1a2, a1b2 - b1a2)

### Solved Examples

Example 1: Find the magnitude of the vector $\stackrel{\to }{a}$$\overrightarrow a$ = 5i - 3j + k, using the formula from vector algebra.

Solution:

The given vector is $\stackrel{\to }{a}$$\overrightarrow a$ = 5i - 3j + k.

The magnitude of the vector is |a| = $\sqrt{{5}^{2}+\left(-3{\right)}^{2}+{1}^{2}}=\sqrt{25+9+1}=\sqrt{35}$$\sqrt{5^2 + (-3)^2 + 1^2} = \sqrt{25 + 9 + 1} = \sqrt{35}$

Therefore, the magnitude of the vector is $\sqrt{35}$$\sqrt{35}$.

Example 2: Using vector algebra concepts, find the dot product between the two vectors 2i + 3j + k, and 5i -2j + 3k.

Solution:

The two given vectors are:

$\stackrel{\to }{a}$$\overrightarrow a$ = 2i + 3i + k, and $\stackrel{\to }{b}$$\overrightarrow b$ = 5i -2j + 3k

Using the dot product we have $\stackrel{\to }{a}.\stackrel{\to }{b}$$\overrightarrow a.\overrightarrow b$ = 2.(5) + 3.(-2) + 1.(3) = 10 - 6 + 3 = 7

Therefore, the dot product of the two vectors is 7.

Solved Problem of Previous year Question Paper

Question 1: Let b = 4i + 3j and c be two vectors perpendicular to each other in the xy-plane. All vectors in the same plane having projections 1 and 2 along b and c respectively are given by _________.

Solution:

Let r = λb + μc and c = ± (xi + yj).

Since c and b are perpendicular, we have 4x + 3y = 0

⇒ c = ±x (i − 43j), {Because, y = $\left[$$[$−4 / 3$\right]$$]$x}

Now, projection of r on b = $\left[$$[$r.b$\right]$$]$ /$\left[$$[$|b|$\right]$$]$ = 1

$\left[$$[$(λb + μc) . b$\right]$$]$ / $\left[$$[$|b|$\right]$$]$

= $\left[$$[$λb . B$\right]$$]$ / $\left[$$[$|b|$\right]$$]$ = 1

⇒ λ = 1 / 5

Again, projection of r on c = $\left[$$[$r.c$\right]$$]$ / $\left[$$[$|c|$\right]$$]$ = 2

This gives μx = $\left[$$[$6 / 5$\right]$$]$

⇒ r = $\left[$$[$1 / 5$\right]$$]$ (4i + 3j) + $\left[$$[$6 / 5$\right]$$]$ (i − $\left[$$[$4 / 3$\right]$$]$j)

= 2i−j or

r = $\left[$$[$1 / 5$\right]$$]$ (4i + 3j) − $\left[$$[$6 / 5$\right]$$]$ (i − $\left[$$[$4 / 3$\right]$$]$j)

= $\left[$$[$−2 / 5$\right]$$]$ i + $\left[$$[$11 / 5$\right]$$]$ j

Question 2: If a, b and c are unit vectors, then |a − b|2 + |b − c|2 + |c − a|2 does not exceed

A) 4

B) 9

C) 8

D) 6

Solution:

|a − b|2 + |b − c|2 + |c − a|2 = 2 (a2 + b2 + c2) − 2 (a $×$$\times$ b + b $×$$\times$ c + c $×$$\times$ a)

= 2 $×$$\times$ 3 − 2 (a $×$$\times$ b + b $×$$\times$ c + c $×$$\times$ a)

= 6 − {(a + b + c)2 − a2− b2 − c2}

= 9 − |a + b + c| 2 ≤ 9

Question 3: Let a, b and c be vectors with magnitudes 3, 4 and 5 respectively and a + b + c = 0, then the values of a . b + b . c + c . a is ________.

Solution:

Since a + b + c = 0

On squaring both sides, we get

|a|2 + |b|2 + |c|2 + 2 (a . b + b . c + c . a) = 0

⇒ 2 (a . b + b . c + c . a) = − (9 + 16 + 25)

⇒ a . b + b . c + c . a = −25

### Practise Questions

Q1. Let $\stackrel{\to }{a}=\stackrel{^}{j}-\stackrel{^}{k}$$\vec{a}=\hat{j}-\hat{k}$ and $\stackrel{\to }{c}=\stackrel{^}{i}-\stackrel{^}{j}-\stackrel{^}{k}.$$\vec{c}=\hat{i}-\hat{j}-\hat{k} .$ Then the vector $\stackrel{\to }{b}$$\vec{b}$ satisfying $\stackrel{\to }{a}×\stackrel{\to }{b}+\stackrel{\to }{c}=\stackrel{\to }{0}$$\vec{a} \times \vec{b}+\vec{c}=\overrightarrow{0}$ and $\stackrel{\to }{a}.\stackrel{\to }{b}=3$$\vec{a} . \vec{b}=3$ is :

A. $-\stackrel{^}{\mathrm{i}}+\stackrel{^}{\mathrm{j}}-2\stackrel{^}{\mathrm{k}}$$-\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$

B. $2\stackrel{^}{\mathrm{i}}-\stackrel{^}{\mathrm{j}}+2\stackrel{^}{\mathrm{k}}$$2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$

C. $\stackrel{^}{\mathrm{i}}-\stackrel{^}{\mathrm{j}}-2\stackrel{^}{\mathrm{k}}$$\hat{\mathrm{i}}-\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$

D. $\stackrel{^}{\mathrm{i}}+\stackrel{^}{\mathrm{j}}-2\stackrel{^}{\mathrm{k}}$$\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$

Q2. If $\stackrel{\to }{\mathrm{a}}=\frac{1}{\sqrt{10}}\left(3\stackrel{^}{\mathrm{i}}+\stackrel{^}{\mathrm{k}}\right)$$\overrightarrow{\mathrm{a}}=\frac{1}{\sqrt{10}}(3 \hat{\mathrm{i}}+\hat{\mathrm{k}})$ and $\stackrel{\to }{\mathrm{b}}=\frac{1}{7}\left(2\stackrel{^}{\mathrm{i}}+3\stackrel{^}{\mathrm{j}}-6\stackrel{^}{\mathrm{k}}\right),$$\overrightarrow{\mathrm{b}}=\frac{1}{7}(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}),$ then the value of $\left(2\stackrel{\to }{\mathrm{a}}-\stackrel{\to }{\mathrm{b}}\right)\cdot \left[\left(\stackrel{\to }{\mathrm{a}}×\stackrel{\to }{\mathrm{b}}\right)×\left(\stackrel{\to }{\mathrm{a}}+2\stackrel{\to }{\mathrm{b}}\right)\right]$$(2 \overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}) \cdot[(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \times(\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}})]$ is 1:

A.  5

B.  3

C.  – 5

D.  – 3

### Conclusion

A vector is defined as an object that has both directions and magnitude in general. In this post, we'll look at several types of vectors, as well as vector algebra operations and applications. Also, for a better grasp of the subject, we went through some of the solved examples and previous year's problems.

See More

## JEE Main Important Dates

View All Dates
JEE Main 2022 June and July Session exam dates and revised schedule have been announced by the NTA. JEE Main 2022 June and July Session will now be conducted on 20-June-2022, and the exam registration closes on 5-Apr-2022. You can check the complete schedule on our site. Furthermore, you can check JEE Main 2022 dates for application, admit card, exam, answer key, result, counselling, etc along with other relevant information.
See More
June
July
View All Dates

## JEE Main Information

Application Form
Eligibility Criteria
Reservation Policy
NTA has announced the JEE Main 2022 June session application form release date on the official website https://jeemain.nta.nic.in/. JEE Main 2022 June and July session Application Form is available on the official website for online registration. Besides JEE Main 2022 June and July session application form release date, learn about the application process, steps to fill the form, how to submit, exam date sheet etc online. Check our website for more details. July Session's details will be updated soon by NTA.

## JEE Main 2022 Study Material

View all study material for JEE Main
JEE Main 2022 Study Materials: Strengthen your fundamentals with exhaustive JEE Main Study Materials. It covers the entire JEE Main syllabus, DPP, PYP with ample objective and subjective solved problems. Free download of JEE Main study material for Physics, Chemistry and Maths are available on our website so that students can gear up their preparation for JEE Main exam 2022 with Vedantu right on time.
See More
All
Mathematics
Physics
Chemistry
Sets, Relations and Functions
Matrices and Determinants
See All

## JEE Main Question Papers

see all
Download JEE Main Question Papers & ​Answer Keys of 2021, 2020, 2019, 2018 and 2017 PDFs. JEE Main Question Paper are provided language-wise along with their answer keys. We also offer JEE Main Sample Question Papers with Answer Keys for Physics, Chemistry and Maths solved by our expert teachers on Vedantu. Downloading the JEE Main Sample Question Papers with solutions will help the engineering aspirants to score high marks in the JEE Main examinations.
See More
PYQP
Sample Paper
2020
2021
2022
2022
January
06th January 2020 - Maths, Physics and Chemistry
English  •   Shift 1
06th January 2020 - Maths, Physics and Chemistry
English  •   Shift 2
07th January 2020 - Maths, Physics and Chemistry
English  •   Shift 1
07th January 2020 - Maths, Physics and Chemistry
English  •   Shift 2
08th January 2020 - Maths, Physics and Chemistry
English  •   Shift 1
08th January 2020 - Maths, Physics and Chemistry
English  •   Shift 2
September
01st September 2020 - Maths, Physics and Chemistry
English  •   Shift 1
01st September 2020 - Maths, Physics and Chemistry
English  •   Shift 2
02nd September 2020 - Maths, Physics and Chemistry
English  •   Shift 1
02nd September 2020 - Maths, Physics and Chemistry
English  •   Shift 2
03rd September 2020 - Maths, Physics and Chemistry
English  •   Shift 1
03rd September 2020 - Maths, Physics and Chemistry
English  •   Shift 2
04th September 2020 - Maths, Physics and Chemistry
English  •   Shift 1
04th September 2020 - Maths, Physics and Chemistry
English  •   Shift 2
05th September 2020 - Maths, Physics and Chemistry
English  •   Shift 1
05th September 2020 - Maths, Physics and Chemistry
English  •   Shift 2
February
23rd February 2021 - Maths, Physics and Chemistry
English  •   Shift 1
23rd February 2021 - Maths, Physics and Chemistry
English  •   Shift 2
24th February 2021 - Maths, Physics and Chemistry
English  •   Shift 1
24th February 2021 - Maths, Physics and Chemistry
English  •   Shift 2
25th February 2021 - Maths, Physics and Chemistry
English  •   Shift 1
25th February 2021 - Maths, Physics and Chemistry
English  •   Shift 2
March
15th March 2021 - Maths, Physics and Chemistry
English  •   Shift 1
15th March 2021 - Maths, Physics and Chemistry
English  •   Shift 2
16th March 2021 - Maths, Physics and Chemistry
English  •   Shift 1
16th March 2021 - Maths, Physics and Chemistry
English  •   Shift 2
17th March 2021 - Maths, Physics and Chemistry
English  •   Shift 1
17th March 2021 - Maths, Physics and Chemistry
English  •   Shift 2
July
19th July 2021 - Maths, Physics and Chemistry
English  •   Shift 1
19th July 2021 - Maths, Physics and Chemistry
English  •   Shift 2
21st July 2021 - Maths, Physics and Chemistry
English  •   Shift 2
24th July 2021 - Maths, Physics and Chemistry
English  •   Shift 1
24th July 2021 - Maths, Physics and Chemistry
English  •   Shift 2
26th July 2021 - Maths, Physics and Chemistry
English  •   Shift 1
August
25th August 2021 - Maths, Physics and Chemistry
English  •   Shift 1
25th August 2021 - Maths, Physics and Chemistry
English  •   Shift 2
26th August 2021 - Maths, Physics and Chemistry
English  •   Shift 1
26th August 2021 - Maths, Physics and Chemistry
English  •   Shift 2
30th August 2021 - aptitude
English  •   Shift 1
30th August 2021 - Maths, Physics and Chemistry
English  •   Shift 2
June
23rd June 2022 - Maths, Physics and Chemistry
English  •   Shift 1
23rd June 2022 - mathematics
English  •   Shift 1
23rd June 2022 - physics
English  •   Shift 1
23rd June 2022 - chemistry
English  •   Shift 1
23rd June 2022 - Maths, Physics and Chemistry
English  •   Shift 2
23rd June 2022 - mathematics
English  •   Shift 2
23rd June 2022 - physics
English  •   Shift 2
23rd June 2022 - chemistry
English  •   Shift 2
24th June 2022 - Maths, Physics and Chemistry
English  •   Shift 1
24th June 2022 - mathematics
English  •   Shift 1
24th June 2022 - physics
English  •   Shift 1
24th June 2022 - chemistry
English  •   Shift 1
24th June 2022 - Maths, Physics and Chemistry
English  •   Shift 2
24th June 2022 - mathematics
English  •   Shift 2
24th June 2022 - physics
English  •   Shift 2
24th June 2022 - chemistry
English  •   Shift 2
25th June 2022 - Maths, Physics and Chemistry
English  •   Shift 1
25th June 2022 - mathematics
English  •   Shift 1
25th June 2022 - physics
English  •   Shift 1
25th June 2022 - chemistry
English  •   Shift 1
25th June 2022 - general
English  •   Shift 2
25th June 2022 - mathematics
English  •   Shift 2
25th June 2022 - physics
English  •   Shift 2
25th June 2022 - chemistry
English  •   Shift 2
26th June 2022 - general
English  •   Shift 1
26th June 2022 - mathematics
English  •   Shift 1
26th June 2022 - physics
English  •   Shift 1
26th June 2022 - chemistry
English  •   Shift 1
26th June 2022 - exam-strategy
English  •   Shift 2
26th June 2022 - mathematics
English  •   Shift 2
26th June 2022 - physics
English  •   Shift 2
26th June 2022 - chemistry
English  •   Shift 2
27th June 2022 - exam-strategy
English  •   Shift 1
27th June 2022 - mathematics
English  •   Shift 1
27th June 2022 - physics
English  •   Shift 1
27th June 2022 - chemistry
English  •   Shift 1
27th June 2022 - general
English  •   Shift 2
27th June 2022 - mathematics
English  •   Shift 2
27th June 2022 - physics
English  •   Shift 2
27th June 2022 - chemistry
English  •   Shift 2
28th June 2022 - general
English  •   Shift 1
28th June 2022 - mathematics
English  •   Shift 1
28th June 2022 - physics
English  •   Shift 1
28th June 2022 - chemistry
English  •   Shift 1
28th June 2022 - general
English  •   Shift 2
28th June 2022 - mathematics
English  •   Shift 2
28th June 2022 - physics
English  •   Shift 2
28th June 2022 - chemistry
English  •   Shift 2
23rd June 2022 - Maths, Physics and Chemistry
English  •   Shift 1
23rd June 2022 - Maths, Physics and Chemistry
English  •   Shift 1
23rd June 2022 - mathematics
English  •   Shift 1
23rd June 2022 - physics
English  •   Shift 1
23rd June 2022 - chemistry
English  •   Shift 1
23rd June 2022 - Maths, Physics and Chemistry
English  •   Shift 2
23rd June 2022 - mathematics
English  •   Shift 2
23rd June 2022 - physics
English  •   Shift 2
23rd June 2022 - chemistry
English  •   Shift 2
24th June 2022 - Maths, Physics and Chemistry
English  •   Shift 1
24th June 2022 - Maths, Physics and Chemistry
English  •   Shift 1
24th June 2022 - mathematics
English  •   Shift 1
24th June 2022 - physics
English  •   Shift 1
24th June 2022 - chemistry
English  •   Shift 1
24th June 2022 - Maths, Physics and Chemistry
English  •   Shift 2
24th June 2022 - mathematics
English  •   Shift 2
24th June 2022 - physics
English  •   Shift 2
24th June 2022 - chemistry
English  •   Shift 2
25th June 2022 - Maths, Physics and Chemistry
English  •   Shift 1
25th June 2022 - Maths, Physics and Chemistry
English  •   Shift 1
25th June 2022 - mathematics
English  •   Shift 1
25th June 2022 - physics
English  •   Shift 1
25th June 2022 - chemistry
English  •   Shift 1
25th June 2022 - general
English  •   Shift 2
25th June 2022 - mathematics
English  •   Shift 2
25th June 2022 - physics
English  •   Shift 2
25th June 2022 - chemistry
English  •   Shift 2
26th June 2022 - general
English  •   Shift 1
26th June 2022 - general
English  •   Shift 1
26th June 2022 - mathematics
English  •   Shift 1
26th June 2022 - physics
English  •   Shift 1
26th June 2022 - chemistry
English  •   Shift 1
26th June 2022 - exam-strategy
English  •   Shift 2
26th June 2022 - mathematics
English  •   Shift 2
26th June 2022 - physics
English  •   Shift 2
26th June 2022 - chemistry
English  •   Shift 2
27th June 2022 - exam-strategy
English  •   Shift 1
27th June 2022 - exam-strategy
English  •   Shift 1
27th June 2022 - mathematics
English  •   Shift 1
27th June 2022 - physics
English  •   Shift 1
27th June 2022 - chemistry
English  •   Shift 1
27th June 2022 - general
English  •   Shift 2
27th June 2022 - mathematics
English  •   Shift 2
27th June 2022 - physics
English  •   Shift 2
27th June 2022 - chemistry
English  •   Shift 2
28th June 2022 - general
English  •   Shift 1
28th June 2022 - general
English  •   Shift 1
28th June 2022 - mathematics
English  •   Shift 1
28th June 2022 - physics
English  •   Shift 1
28th June 2022 - chemistry
English  •   Shift 1
28th June 2022 - general
English  •   Shift 2
28th June 2022 - mathematics
English  •   Shift 2
28th June 2022 - physics
English  •   Shift 2
28th June 2022 - chemistry
English  •   Shift 2
July
24th July 2022 - Maths, Physics and Chemistry
English  •   Shift 1, 2
24th July 2022 - Maths, Physics and Chemistry
English  •   Shift 1
24th July 2022 - mathematics
English  •   Shift 1
24th July 2022 - physics
English  •   Shift 1
24th July 2022 - chemistry
English  •   Shift 1
24th July 2022 - Maths, Physics and Chemistry
English  •   Shift 2
24th July 2022 - mathematics
English  •   Shift 2
24th July 2022 - physics
English  •   Shift 2
24th July 2022 - chemistry
English  •   Shift 2
25th July 2022 - Maths, Physics and Chemistry
English  •   Shift 1, 2
25th July 2022 - Maths, Physics and Chemistry
English  •   Shift 1
25th July 2022 - mathematics
English  •   Shift 1
25th July 2022 - physics
English  •   Shift 1
25th July 2022 - chemistry
English  •   Shift 1
25th July 2022 - Maths, Physics and Chemistry
English  •   Shift 2
25th July 2022 - mathematics
English  •   Shift 2
25th July 2022 - physics
English  •   Shift 2
25th July 2022 - chemistry
English  •   Shift 2
26th July 2022 - Maths, Physics and Chemistry
English  •   Shift 1, 2
26th July 2022 - Maths, Physics and Chemistry
English  •   Shift 1
26th July 2022 - mathematics
English  •   Shift 1
26th July 2022 - physics
English  •   Shift 1
26th July 2022 - chemistry
English  •   Shift 1
26th July 2022 - Maths, Physics and Chemistry
English  •   Shift 2
26th July 2022 - mathematics
English  •   Shift 2
26th July 2022 - physics
English  •   Shift 2
26th July 2022 - chemistry
English  •   Shift 2
27th July 2022 - Maths, Physics and Chemistry
English  •   Shift 1, 2
27th July 2022 - Maths, Physics and Chemistry
English  •   Shift 1
27th July 2022 - mathematics
English  •   Shift 1
27th July 2022 - physics
English  •   Shift 1
27th July 2022 - chemistry
English  •   Shift 1
27th July 2022 - Maths, Physics and Chemistry
English  •   Shift 2
27th July 2022 - mathematics
English  •   Shift 2
27th July 2022 - physics
English  •   Shift 2
27th July 2022 - chemistry
English  •   Shift 2
28th July 2022 - Maths, Physics and Chemistry
English  •   Shift 1, 2
28th July 2022 - Maths, Physics and Chemistry
English  •   Shift 1
28th July 2022 - mathematics
English  •   Shift 1
28th July 2022 - physics
English  •   Shift 1
28th July 2022 - chemistry
English  •   Shift 1
28th July 2022 - Maths, Physics and Chemistry
English  •   Shift 2
28th July 2022 - mathematics
English  •   Shift 2
28th July 2022 - physics
English  •   Shift 2
28th July 2022 - chemistry
English  •   Shift 2
June, July
mathematics
English  •   Shift 1, 2
mathematics
English  •   Shift 1, 2
mathematics
English  •   Shift 1, 2
mathematics
English  •   Shift 1, 2
mathematics
English  •   Shift 1, 2
chemistry
English  •   Shift 1, 2
mathematics
English  •   Shift 1, 2

View all JEE Main Important Books
In order to prepare for JEE Main 2022, candidates should know the list of important books i.e. RD Sharma Solutions, NCERT Solutions, RS Aggarwal Solutions, HC Verma books and RS Aggarwal Solutions. They will find the high quality readymade solutions of these books on Vedantu. These books will help them in order to prepare well for the JEE Main 2022 exam so that they can grab the top rank in the all India entrance exam.
See More
Maths
NCERT Book for Class 12 Maths
Physics
NCERT Book for Class 12 Physics
See All

## JEE Main Mock Tests

View all mock tests
JEE Main 2022 free online mock test series for exam preparation are available on the Vedantu website for free download. Practising these mock test papers of Physics, Chemistry and Maths prepared by expert teachers at Vedantu will help you to boost your confidence to face the JEE Main 2022 examination without any worries. The JEE Main test series for Physics, Chemistry and Maths that is based on the latest syllabus of JEE Main and also the Previous Year Question Papers.
See More
JEE MAIN MOCK TEST - 1
3 hr  • 75 questions • OBJECTIVE
JEE MAIN MOCK TEST - 3
3 hr  • 75 questions • OBJECTIVE
JEE MAIN MOCK TEST - 2
3 hr  • 75 questions • OBJECTIVE

## Master Teachers

From IITs & other top-tier colleges with 5+ years of experience
You can count on our specially-trained teachers to bring out the best in every student.
They have taught over 4.5 crore hours to 10 lakh students in 1000+ cities in 57 countries
11+ years exp

### Shreyas

Physics master teacher

4+ years exp

### Nidhi Sharma

Chemistry master teacher

2+ years exp

### Luv Mehan

Chemistry Master Teacher

## JEE Main 2022 Cut-Off

JEE Main Cut Off
NTA is responsible for the release of the JEE Main 2022 June and July Session cut off score. The qualifying percentile score might remain the same for different categories. According to the latest trends, the expected cut off mark for JEE Main 2022 June and July Session is 50% for general category candidates, 45% for physically challenged candidates, and 40% for candidates from reserved categories. For the general category, JEE Main qualifying marks for 2021 ranged from 87.8992241 for general-category, while for OBC/SC/ST categories, they ranged from 68.0234447 for OBC, 46.8825338 for SC and 34.6728999 for ST category.
See More

## JEE Main 2022 Results

JEE Main 2022 June and July Session Result - NTA has announced JEE Main result on their website. To download the Scorecard for JEE Main 2022 June and July Session, visit the official website of JEE Main NTA.
See More
Rank List
Counselling
Cutoff
JEE Main 2022 state rank lists will be released by the state counselling committees for admissions to the 85% state quota and to all seats in NITs and CFTIs colleges. JEE Main 2022 state rank lists are based on the marks obtained in entrance exams. Candidates can check the JEE Main 2022 state rank list on the official website or on our site.

## JEE Top Colleges

View all JEE Main 2022 Top Colleges
Want to know which Engineering colleges in India accept the JEE Main 2022 scores for admission to Engineering? Find the list of Engineering colleges accepting JEE Main scores in India, compiled by Vedantu. There are 1622 Colleges that are accepting JEE Main. Also find more details on Fees, Ranking, Admission, and Placement.
See More

## Counselling

Need more details? Our expert academic counsellors will be happy to patiently explain everything that you want to know.
Speak to an expert

## FAQs on JEE Chapter - Vector Algebra

FAQ

1. What are the objectives of learning about the Vector Product of Two Vectors?

There are several learning objectives for the Vector Product of Two Vectors, which can be supplied as follows:

• It explains the difference between the scalar product and the vector product, which is formed by multiplying two vectors.

• It aids in determining the product of two vectors and, as a result, determining whether they can be joined or not.

• Students will also learn how vectors are useful in physics and in a variety of calculations.

2. How to prepare Vector Algebra?

One of the essential topics is vector algebra, which you can prepare for by mastering a few basic concepts:

• Beginning with a basic understanding of vectors and all of the words used in vector algebra.

• The representation of a vector is a crucial aspect of this chapter. It is critical that you read all of the questions slowly and deliberately.

• Because vectors are all about magnitude and direction, double-check that the direction specified in the question and the direction acquired in the solution are correct.

• After studying certain sections/ideas, be sure you solve questions relating to those concepts without consulting the solutions, practise MCQ from the above-mentioned books, and solve all of the previous year's JEE problems.

3. What are the uses of vector algebra in the physical world?

Using the notion of vector algebra, the physical quantities of displacement, velocity, position, force, and torque are all represented in a three-dimensional plane. In addition, vector algebra allows for a variety of addition and multiplication operations on these numbers.

## JEE Main Upcoming Dates

Vedantu offers free live Master Classes for CBSE Class 6 to 12, ICSE, JEE Main, JEE 2022, & more by India’s best teachers. Learn all the important concepts concisely along with amazing tricks to score high marks in your class and other competitive exams.
See More