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Class 12 Maths Chapter 10: Vector Algebra

Vector Algebra introduces mathematical quantities that possess both magnitude and direction. Unlike scalars, vectors are widely used to represent physical quantities such as displacement, velocity, acceleration, and force. This chapter covers vector operations, position vectors, scalar (dot) product, vector (cross) product, and their geometric interpretations. It lays the foundation for Three-Dimensional Geometry and is an important scoring chapter in CBSE Board, JEE Main, and JEE Advanced.

At ALLEN, Class 12 Maths Chapter 10: Vector Algebra smart study resources contains concise concepts, important formulas, graphical interpretation, shortcut techniques, and exam-oriented practice. These notes help students strengthen their understanding and solve vector-based problems accurately and efficiently.

1.0Chapter Snapshot

Parameter	Details
Difficulty Level	Moderate
Revision Time	3–4 Hours
Key Concepts Covered	Vector Operations, Dot Product, Cross Product, Angle Between Vectors

2.0Chapter 10 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution
Practice Test Paper with Answers

Test your Knowledge

question 1 of 5

A quantity having both magnitude and direction is called a:

1.Scalar

2.Vector

3.Constant

4.Variable

Must-Revise Concepts

Angle Between Two Vectors

Collinear Vectors

Components of a Vector

Coplanar Vectors

Dot Product

Dot Product of Two Vectors

Dot Product of Two Vectors Questions

Eigenvectors of a Matrix

Multiplication of a Vector

Negative of a vector

Scalar triple product

Types of Vectors

Vector Algebra

Vector Algebra PYQ with solutions

Vector Triple Product

Vectors in Maths

What are Vector and Scalar Quantities

Vector Algebra

Vector Algebra is a branch of mathematics that deals with quantities having both magnitude and direction. Unlike simple numbers, vectors can represent more complex entities such as force, velocity, and displacement.

In Vector Algebra, we use arrows to visually represent vectors. The length of the arrow indicates the vector's magnitude, while the direction of the arrow shows its direction. This graphical representation helps in understanding and solving problems in physics, engineering, and computer science.

Key operations in Vector Algebra include addition, subtraction, and multiplication of vectors. These operations follow specific rules, such as the Triangle Law and Parallelogram Law, which help in combining vectors and analyzing their interactions.

In this article, we will explore the essential concepts of Vector Algebra, including the definition and properties of vectors, key operations such as addition, subtraction, and multiplication, and the practical applications of vectors in fields like physics, engineering, and computer graphics. By the end, you'll have a solid understanding of how vectors are used to model and solve real-world problems efficiently.

1.0What is Vector Algebra

Vector Algebra is a branch of mathematics that deals with vectors, which are mathematical objects that have both magnitude (size) and direction. In vector algebra, vectors are represented as directed line segments or arrows in space. They are used to represent quantities such as force, velocity, acceleration, and displacement in physics, as well as other concepts in various fields like engineering, computer graphics, and economics.

Vector algebra involves operations such as addition, subtraction, and multiplication of vectors, which follow specific rules and properties. These operations allow for the manipulation and analysis of vectors to solve problems and describe physical phenomena. Key concepts in vector algebra include vector components, dot product, cross product, vector spaces, and applications in geometry, mechanics, and other areas of mathematics and science.

2.0Types of Vectors in Vector Algebra

In vector algebra, vectors can be classified into various types based on their properties and roles in mathematical operations. Here are the main types of vectors:

Zero Vector (Null Vector)

Definition: A vector with a magnitude of zero and no specific direction.

Notation: 0

Example: $0 = 0 \hat{i} + 0 \hat{j} + 0 \hat{k}$

Unit Vector

Definition: A vector with a magnitude of one, used to indicate direction.

Notation: $\hat{A}$

Example: If $A = 3 \hat{i} + 4 \hat{j}$ , then $A = \frac{A}{∣ A ∣} = \frac{3 i + 4 j}{5}$

Position Vector

Definition: A vector that represents the position of a point relative to the origin.

Notation: r

Example: The position vector of a point P (x, y, z) is $r = x \hat{i} + y \hat{j} + z \hat{k}$

Co-initial Vectors

Definition: Vectors that have the same initial point.

Example: Vectors A and B starting from the same point O.

Collinear Vectors

Definition: Vectors that lie along the same line or are parallel to each other.

Example: Vectors $A = 2 \hat{i} + 3 \hat{j}$

and $B = 4 \hat{i} + 6 \hat{j}$ are collinear.

Equal Vectors

Definition: Vectors that have the same magnitude and direction, regardless of their initial points.

Example: $A = 3 \hat{i} + 4 \hat{j}$ and $B = 3 \hat{i} + 4 \hat{j}$

Negative of a Vector

Definition: A vector that has the same magnitude as a given vector but points in the opposite direction.

Example: If $A = 5 \hat{i} - 2 \hat{j}$ , then $- A = - 5 \hat{i} + 2 \hat{j}$

Parallel Vectors

Definition: Vectors that have the same or opposite direction.

Example: Vectors A and KA (where k is a scalar) are parallel.

Orthogonal (Perpendicular) Vectors

Definition: Vectors that are at right angles (90 degrees) to each other.

Example: If $A = \hat{i} + 2 \hat{j}$ and $B = - 2 \hat{i} + \hat{j}$ , then A. B = 0

Coplanar Vectors

Definition: Vectors that lie in the same plane.

Example: Vectors $A = 2 \hat{i} + 3 \hat{j}$ and B= $- \hat{i} + 4 \hat{j}$ are coplanar with any vector in the xy-plane.

These different types of vectors help in classifying and understanding their properties and how they interact with each other in various mathematical and physical contexts.

Related Video:

1.0Vector Algebra Operations

Vector Algebra Operations encompass a variety of mathematical manipulations involving vectors. These operations are fundamental in understanding and analyzing vectors in various fields such as physics, engineering, and computer science. Here are the primary Vector Algebra Operations:

Vector Addition

Let $a$ and $b$ be two vectors in a plane, which are represented by $AB$ and $CD$ . Their addition can be performed in the following two ways.

Triangle Law of Addition of Vectors

If two vectors are represented in magnitude & direction by two sides of a triangle taken in same order, then their sum is represented by the third side taken in reverse order.

Parallelogram Law of Addition of Vectors

If two vectors be represented in magnitude and direction by the two adjacent sides of a parallelogram then their sum will be represented by the diagonal through the co-initial point.

Vector Subtraction

Vector $- b$ has length equals to vector $b$ but its direction is opposite. Subtraction of vector $a$ and $b$ is defined as addition of $a$ and ( $- b$ ). It is written as follows:

$a - b = a + (- b)$

Scalar Multiplication or Scalar Product

Let $a$ and $b$ be two non-zero vectors inclined at an angle θ. Then the scalar product of $a$ with $b$ is denoted by $a \cdot b$ and is defined as $a \cdot b = ∣ a ∣∣ b ∣ cos θ; 0 \leq θ \leq π$ .

$a \cdot b = ∣ a ∥ b ∣ cos θ (0 \leq θ \leq π)$

Note: if θ is acute then $a \cdot b > 0$ & if θ is obtuse then $a \cdot b < 0$

(i) $a \cdot a = ∣ a ∣^{2} = a^{2}$

(ii) $a \cdot b = b \cdot a (co mm u t a t i v e)$

$a \cdot (b + c) = a \cdot b + a \cdot c$ (distributive)
$a \cdot b = 0 \Leftrightarrow a ⊥ b; (a, b \neq = 0)$
$\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1; \hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$

Linear Combination

A vector $r$ is said to Linear Combination of vectors $a, b, c, \dots$ if ∃ scalars x, y, z such that

$r = x a + y b + z c$

Dot Product

If $a = a_{1} \hat{i} + 2 a \hat{j} + a_{3} \hat{k} & = b = b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k}$ then $a \cdot b$ = a₁b₁ + a₂b₂+ a₃b₃ &

$cos θ = \frac{a \cdot b =}{∣ a ∣∣ b ∣} = \frac{a _{1} b _{1} + a _{2} b _{2} + a _{3} b _{3}}{a _{1}^{2} + a _{2}^{2} + a _{3}^{2} \cdot b _{1}^{2} + b _{2}^{2} + b _{3}^{2}}$

Cross Product

The vector product of two nonzero vectors $a$ and $b$ , is denoted by $a \times b$ and defined as $a \times b = ∣ a ∥ b ∣ sin θ \overset{n}{^}$

Where, θ is the angle between $a$ and $b$ , 0 ≤ θ ≤ π and $\overset{n}{^}$ is a unit vector perpendicular to both $a$ and $b$ , such that $a$ , $b$ and $\hat{n}$ form a right handed system. i.e., the right handed system rotated from $a$ to $b$ moves in the direction of $\overset{n}{^}$ .

If either $a = 0 or b = 0$ , then θ is not defined and in this case , we define $a \times b = 0$ .

2.0Vector Algebra Formulas

Vector Addition

$A + B = (A_{x} + B_{x}) \hat{i} + (A_{y} + B_{y}) \hat{j} + (A_{z} + B_{z}) \hat{k}$

Combine corresponding components of vectors A and B.

Vector Subtraction

$A - B = (A_{x} - B_{x}) \hat{i} + (A_{y} - B_{y}) \hat{j} + (A_{z} - B_{z}) \hat{k}$

Subtract corresponding components of vector B and A.

Scalar Multiplication

$k A = (k A_{x}) \hat{i} + (k A_{y}) \hat{j} + (k A_{z}) \hat{k}$

Multiply each component of vector A by the scalar k.

Dot Product

$A \cdot B = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}$

Calculate the sum of the products of the corresponding components of vectors A and B.

Alternatively, using magnitudes and the cosine of the angle θ between the vectors:

$A \cdot B = ∣ A ∥ B ∣ cos θ$

Note: If $A$ and $B$ are Parallel then $A \times B$ are Perpendicular then $A \cdot B = 0$

Cross Product

$A \times B = (A_{y} B_{z} - A_{z} B_{y}) \hat{i} - (A_{x} B_{z} - A_{z} B_{x}) \hat{j} + (A_{x} B_{y} - A_{y} B_{x}) \hat{k}$

Compute a vector perpendicular to both A and B with magnitude equal to the area of the parallelogram they span.

Vector Magnitude

$∣ A ∣ = A_{x}^{2} + A_{y}^{2} + A_{z}^{2}$

Calculate the length of vector A using its components.

Unit Vector

$A = \frac{A}{∣ A ∣}$

Divide vector A by its magnitude to get a unit vector in the same direction.

Projection of A onto B

$proj_{B} A = (\frac{A \cdot B}{∣ B ∣ ^{2}}) B$

Project vector A onto B, resulting in a vector parallel to B.

Angle Between Two Vectors

$cos θ = \frac{A \cdot B}{∣ A ∥ B ∣}$

Find the cosine of the angle between vectors A and B.

3.0Vector Algebra Examples

Example 1: Find unit vector of $A = 3 \hat{i} + 4 \hat{j} - 5 \hat{k}$

Solution:

$∣ A ∣ = 9 + 16 + 25 = 50$ ⇒ unit vector: $\frac{3 i ^ + 4 j ^ - 5 k ^}{50}$

Example 2: If $A \equiv (2 \hat{i} + 3 \hat{j}), B \equiv (p \hat{i} + 9 \hat{j})$ and $C \equiv (\hat{i} - \hat{j})$ are collinear, then the value of p is:

(A) 1/2 (B) 3/2 (C) 7/2 (D) 5/2

Ans. (C)

Solution:

$AB = (p - 2) \hat{i} + 6 \hat{j}, AC = - \hat{i} - 4 \hat{j}$

Now A, B, C are collinear ⇔ $A B ∥ A C$ ⇔ $\frac{p - 2}{- 1} = \frac{6}{- 4}$ ⇔ $p = \frac{7}{2}$

Example 3: The value of λ when $a = 2 \hat{i} - 3 \hat{j} + \hat{k}$ and $b = 8 \hat{i} + λ \hat{j} + 4 \hat{k}$ are parallel is:

(A) 4 (B) –6 (C) –12 (D) 1

Ans. (C)

Solution:

Since $a$ & $b$ are parallel ⇒ $\frac{2}{8}$ = $- \frac{3}{λ}$ = $\frac{1}{4}$ ⇒ –12

Example 4: If $a$ and $b$ are unit vectors, then which of the following values of $a \cdot b$ is not possible?

(A) $3$ (B) $\frac{3}{2}$

(C) $\frac{1}{2}$ (D) $\frac{- 1}{2}$

Ans. (A)

Solution:

Since $a \cdot b = ∣ a ∣∣ b ∣ cos θ = cos θ$ and cosθ never be equal to $3$

Example 5: If p^th, q^th, r^th terms of a G.P. are the positive numbers a, b, c then angle between the vectors $lo g a^{2} \hat{i} + lo g b^{2} \hat{j} + lo g c^{2} \hat{k}$ and (q-r) $\hat{i} + (r - p) \hat{j} + (p - q) \hat{k}$ is:

(A) $\frac{p}{3}$

(B) $\frac{p}{2}$

(C) $sin^{- 1} (\frac{1}{a ^{2} + b ^{2} + c ^{2}})$

(D) none of these

Ans. (B)

Solution:

Let x₀ be first term and x the common ratio of the G.P.

∴ a = x₀ x^p-1, b = x₀x^q–1, c = x₀x^r-1

⇒ log a = log x₀ + (p–1) log x ; log b = log x₀ +(q–1) log x; log c = log x₀ +(r–1) log x

If $a = lo g a^{2} \hat{i} + lo g b^{2} \hat{j} + lo g c^{2} \hat{k}$ and $b = (q - r) \hat{i} + (r - p) \hat{j} + (p - q) \hat{k}$

$∴ a \cdot b = Σ2 (lo g a) (q - r) = 2 \sum (lo g x_{0} + (p - 1) lo g x) (q - r) = 0$

⇒ $a^{\land} b = \frac{π}{2} .$

Example 6: A unit vector perpendicular to the plane determined by the points (1,–1,2), (2,0,–1) and (0,2,1) is

(A) $\pm \frac{1}{6} (2 \hat{i} + \hat{j} + \hat{k})$

(B) $\frac{1}{6} (2 \hat{i} + \hat{j} + \hat{k})$

(C) $\frac{1}{6} (\hat{i} + \hat{j} + \hat{k})$

(D) $\frac{1}{6} (2 \hat{i} - \hat{j} - \hat{k})$

Ans. (A)

Solution:

$a = \hat{i} + \hat{j} - 3 \hat{k}, b = - 2 \hat{i} + 2 \hat{j} + 2 \hat{k}$

$a \times b = \hat{i} 1 - 2 \hat{j} 12 \hat{k} - 3 2 = 8 \hat{i} + 4 \hat{j} + 4 \hat{k}$

Hence unit vector = $\pm \frac{2 i ^ + j ^ + k ^}{6}$

Example 7: If $a = 2 \hat{i} + 3 \hat{j} - 5 \hat{k}, b = m \hat{i} + n \hat{j} + 12 \hat{k}$ and $a \times b = 0$ then (m,n)=

(A) $(- \frac{24}{5}, \frac{36}{5})$ (B) $(\frac{24}{5}, - \frac{36}{5})$

(C) $(- \frac{24}{5}, - \frac{36}{5})$ (D) $(\frac{24}{5}, \frac{36}{5})$

Ans. (C)

Solution:

$a \times b = \hat{i} 2 m \hat{j} 3 n \hat{k} - 5 12$ = $(36 + 5 n) \hat{i} - (24 + 5 m) \hat{j} + (2 n - 3 m) \hat{k}$ =0

$\Rightarrow m = \frac{- 24}{5}, n = \frac{- 36}{5}$

4.0Solved Questions on Vector Algebra

What is a unit vector?

Ans: A unit vector is a vector with a magnitude of one. It is used to indicate direction. If A is any vector, its unit vector is $A = \frac{A}{∣ A ∣}$ , where $∣ A ∣$ is the magnitude of A.

How do you find the magnitude of a vector?

Ans: The magnitude of a vector $A = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}$ is calculated as $∣ A ∣ = A_{x}^{2} + A_{y}^{2} + A_{z}^{2}$

What is the projection of one vector onto another?

Ans: The projection of vector A onto vector B is given by $proj_{B} A = (\frac{A \cdot B}{∣ B ∣ ^{2}}) B$ . It represents the component of A in the direction of B.

What is a position vector?

Ans: A position vector is a vector that represents the position of a point in space relative to an origin. For a point P(x, y, z), the position vector is $r = x \hat{i} + y \hat{j} + z \hat{k}$ .

How do you subtract one vector from another?

Ans: To subtract vector B from vector A, you subtract their corresponding components:

$A - B = (A_{x} - B_{x}) \hat{i} + (A_{y} - B_{y}) \hat{j} + (A_{z} - B_{z}) \hat{k}$

Also Read:

Coplanar Vectors	Vector Triple Product	Components of a Vector
Scalar Triple Product	Vector Algebra Previous Year Questions with Solutions	Multiplication of a Vector
Vectors in Maths	Collinear Vectors	Angle Between Two Vectors

1.0Learning Outcomes

After completing this chapter, you will be able to:

Understand the concept of vectors and scalars.
Perform vector addition and subtraction.
Represent vectors in component form.
Calculate the magnitude and direction of vectors.
Apply scalar and vector products.
Find the angle and projection between vectors.
Solve JEE Main, JEE Advanced, and CBSE Board questions confidently.

2.05-Minute Quick Revision

Must-Revise Concepts

Scalars and Vectors
Position Vector
Magnitude of a Vector
Unit Vector
Addition and Subtraction of Vectors
Direction Cosines
Section Formula
Scalar (Dot) Product
Vector (Cross) Product
Projection of a Vector

Important Formula Recall

Magnitude: $∣ a ∣ = a_{x}^{2} + a_{y}^{2} + a_{z}^{2}$
Unit Vector: $\overset{a}{^} = \frac{a}{∣ a ∣}$
Dot Product: $a \cdot b = ∣ a ∣∣ b ∣ cos θ$
Cross Product: $∣ a \times b ∣ = ∣ a ∣∣ b ∣ sin θ$

Important Properties / Shortcut Tips

Use the dot product to determine the angle between vectors.
The dot product is zero for perpendicular vectors.
The cross product is zero for parallel vectors.
Always simplify vectors into component form before calculations.
Normalize vectors before finding projections.

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Vector Operations	High	Easy–Moderate
Position & Unit Vectors	High	Easy
Scalar (Dot) Product	Very High	Moderate
Vector (Cross) Product	Very High	Moderate
Projection of Vectors	High	Moderate
Angle Between Vectors	Very High	Moderate
Mixed Vector Problems	Very High	High

4.0Formula Handbook

Available soon

5.0Common Mistakes & JEE Tips

Common Mistake	Correct Approach
Confusing dot product with cross product	Use the dot product for angles and the cross product for perpendicular vectors and area.
Incorrect vector components	Express vectors in component form before calculations.
Forgetting unit vectors	Normalize vectors while finding direction or projection.
Sign errors during vector addition	Add components carefully along each axis.
Wrong angle formula	Apply the dot product formula while calculating the angle between vectors.
Ignoring vector direction	Consider both magnitude and direction in every operation.
Algebraic mistakes	Simplify expressions before performing vector operations.

6.0ALLEN Faculty Tips

Practice vector operations in component form regularly.
Memorize the properties of dot and cross products.
Learn the geometric interpretation of vector products.
Revise projection and angle formulas frequently.
Strengthen vector concepts before studying Three-Dimensional Geometry.
Solve Previous Years' Questions (PYQs) for faster problem-solving.

7.0PYQ Trend Analysis

Vector Algebra is one of the most scoring chapters in JEE Main, JEE Advanced, and CBSE Board examinations. Questions are commonly asked on vector operations, dot product, cross product, projection, and geometric applications.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Vector Operations	1	Easy
Dot Product	1–2	Moderate
Cross Product	1–2	Moderate
Projection & Direction	1	Moderate
Mixed Vector Problems	1–2	Moderate–High

8.0Most Common Question Types

Finding the magnitude and direction of vectors.
Solving vector addition and subtraction problems.
Calculating the angle between two vectors.
Applying scalar and vector products.
Finding the projection of one vector on another.
Determining perpendicular or parallel vectors.
Solving mixed vector algebra problems.
Concept-based and previous years' examination questions.

9.0Smart Revision Strategy

Revise vector notation and component form.
Practice magnitude and unit vector problems.
Learn dot product applications.
Revise cross product properties.
Practice angle and projection questions.
Solve mixed vector algebra problems.
Practice Previous Years' Questions (PYQs).
Complete revision with a timed mock test.

Class 12 Maths Chapters List:- JEE Revision Notes

Chapter 1 Relations and Functions

Chapter 2 Inverse Trigonometric Functions

Chapter 3 Matrices

Chapter 4 Determinants

Chapter 5 Continuity and Differentiability

Chapter 6 Application of Derivatives

Chapter 7 Integrals

Chapter 8 Application of Integrals

Chapter 9 Differential Equations

Chapter 10 Vector Algebra

Chapter 11 Three Dimensional Geometry

Chapter 12 Linear Programming

Chapter 13 Probability

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Class 12 Maths Chapter 10: Vector Algebra

Vector Algebra introduces mathematical quantities that possess both magnitude and direction. Unlike scalars, vectors are widely used to represent physical quantities such as displacement, velocity, acceleration, and force. This chapter covers vector operations, position vectors, scalar (dot) product, vector (cross) product, and their geometric interpretations. It lays the foundation for Three-Dimensional Geometry and is an important scoring chapter in CBSE Board, JEE Main, and JEE Advanced.

At ALLEN, Class 12 Maths Chapter 10: Vector Algebra smart study resources contains concise concepts, important formulas, graphical interpretation, shortcut techniques, and exam-oriented practice. These notes help students strengthen their understanding and solve vector-based problems accurately and efficiently.

1.0Chapter Snapshot

Parameter	Details
Difficulty Level	Moderate
Revision Time	3–4 Hours
Key Concepts Covered	Vector Operations, Dot Product, Cross Product, Angle Between Vectors

2.0Chapter 10 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution
Practice Test Paper with Answers

Must-Revise Concepts

Angle Between Two Vectors

Collinear Vectors

Components of a Vector

Coplanar Vectors

Dot Product

Dot Product of Two Vectors

Dot Product of Two Vectors Questions

Eigenvectors of a Matrix

Multiplication of a Vector

Negative of a vector

Scalar triple product

Types of Vectors

Vector Algebra

Vector Algebra PYQ with solutions

Vector Triple Product

Vectors in Maths

What are Vector and Scalar Quantities

Vector Algebra

Vector Algebra is a branch of mathematics that deals with quantities having both magnitude and direction. Unlike simple numbers, vectors can represent more complex entities such as force, velocity, and displacement.

In Vector Algebra, we use arrows to visually represent vectors. The length of the arrow indicates the vector's magnitude, while the direction of the arrow shows its direction. This graphical representation helps in understanding and solving problems in physics, engineering, and computer science.

Key operations in Vector Algebra include addition, subtraction, and multiplication of vectors. These operations follow specific rules, such as the Triangle Law and Parallelogram Law, which help in combining vectors and analyzing their interactions.

In this article, we will explore the essential concepts of Vector Algebra, including the definition and properties of vectors, key operations such as addition, subtraction, and multiplication, and the practical applications of vectors in fields like physics, engineering, and computer graphics. By the end, you'll have a solid understanding of how vectors are used to model and solve real-world problems efficiently.

1.0What is Vector Algebra

Vector Algebra is a branch of mathematics that deals with vectors, which are mathematical objects that have both magnitude (size) and direction. In vector algebra, vectors are represented as directed line segments or arrows in space. They are used to represent quantities such as force, velocity, acceleration, and displacement in physics, as well as other concepts in various fields like engineering, computer graphics, and economics.

Vector algebra involves operations such as addition, subtraction, and multiplication of vectors, which follow specific rules and properties. These operations allow for the manipulation and analysis of vectors to solve problems and describe physical phenomena. Key concepts in vector algebra include vector components, dot product, cross product, vector spaces, and applications in geometry, mechanics, and other areas of mathematics and science.

2.0Types of Vectors in Vector Algebra

In vector algebra, vectors can be classified into various types based on their properties and roles in mathematical operations. Here are the main types of vectors:

Zero Vector (Null Vector)

Definition: A vector with a magnitude of zero and no specific direction.

Notation: 0

Example: $0 = 0 \hat{i} + 0 \hat{j} + 0 \hat{k}$

Unit Vector

Definition: A vector with a magnitude of one, used to indicate direction.

Notation: $\hat{A}$

Example: If $A = 3 \hat{i} + 4 \hat{j}$ , then $A = \frac{A}{∣ A ∣} = \frac{3 i + 4 j}{5}$

Position Vector

Definition: A vector that represents the position of a point relative to the origin.

Notation: r

Example: The position vector of a point P (x, y, z) is $r = x \hat{i} + y \hat{j} + z \hat{k}$

Co-initial Vectors

Definition: Vectors that have the same initial point.

Example: Vectors A and B starting from the same point O.

Collinear Vectors

Definition: Vectors that lie along the same line or are parallel to each other.

Example: Vectors $A = 2 \hat{i} + 3 \hat{j}$

and $B = 4 \hat{i} + 6 \hat{j}$ are collinear.

Equal Vectors

Definition: Vectors that have the same magnitude and direction, regardless of their initial points.

Example: $A = 3 \hat{i} + 4 \hat{j}$ and $B = 3 \hat{i} + 4 \hat{j}$

Negative of a Vector

Definition: A vector that has the same magnitude as a given vector but points in the opposite direction.

Example: If $A = 5 \hat{i} - 2 \hat{j}$ , then $- A = - 5 \hat{i} + 2 \hat{j}$

Parallel Vectors

Definition: Vectors that have the same or opposite direction.

Example: Vectors A and KA (where k is a scalar) are parallel.

Orthogonal (Perpendicular) Vectors

Definition: Vectors that are at right angles (90 degrees) to each other.

Example: If $A = \hat{i} + 2 \hat{j}$ and $B = - 2 \hat{i} + \hat{j}$ , then A. B = 0

Coplanar Vectors

Definition: Vectors that lie in the same plane.

Example: Vectors $A = 2 \hat{i} + 3 \hat{j}$ and B= $- \hat{i} + 4 \hat{j}$ are coplanar with any vector in the xy-plane.

These different types of vectors help in classifying and understanding their properties and how they interact with each other in various mathematical and physical contexts.

Related Video:

1.0Vector Algebra Operations

Vector Algebra Operations encompass a variety of mathematical manipulations involving vectors. These operations are fundamental in understanding and analyzing vectors in various fields such as physics, engineering, and computer science. Here are the primary Vector Algebra Operations:

Vector Addition

Let $a$ and $b$ be two vectors in a plane, which are represented by $AB$ and $CD$ . Their addition can be performed in the following two ways.

Triangle Law of Addition of Vectors

If two vectors are represented in magnitude & direction by two sides of a triangle taken in same order, then their sum is represented by the third side taken in reverse order.

Parallelogram Law of Addition of Vectors

If two vectors be represented in magnitude and direction by the two adjacent sides of a parallelogram then their sum will be represented by the diagonal through the co-initial point.

Vector Subtraction

Vector $- b$ has length equals to vector $b$ but its direction is opposite. Subtraction of vector $a$ and $b$ is defined as addition of $a$ and ( $- b$ ). It is written as follows:

$a - b = a + (- b)$

Scalar Multiplication or Scalar Product

Let $a$ and $b$ be two non-zero vectors inclined at an angle θ. Then the scalar product of $a$ with $b$ is denoted by $a \cdot b$ and is defined as $a \cdot b = ∣ a ∣∣ b ∣ cos θ; 0 \leq θ \leq π$ .

$a \cdot b = ∣ a ∥ b ∣ cos θ (0 \leq θ \leq π)$

Note: if θ is acute then $a \cdot b > 0$ & if θ is obtuse then $a \cdot b < 0$

(i) $a \cdot a = ∣ a ∣^{2} = a^{2}$

(ii) $a \cdot b = b \cdot a (co mm u t a t i v e)$

$a \cdot (b + c) = a \cdot b + a \cdot c$ (distributive)
$a \cdot b = 0 \Leftrightarrow a ⊥ b; (a, b \neq = 0)$
$\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1; \hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$

Linear Combination

A vector $r$ is said to Linear Combination of vectors $a, b, c, \dots$ if ∃ scalars x, y, z such that

$r = x a + y b + z c$

Dot Product

If $a = a_{1} \hat{i} + 2 a \hat{j} + a_{3} \hat{k} & = b = b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k}$ then $a \cdot b$ = a₁b₁ + a₂b₂+ a₃b₃ &

$cos θ = \frac{a \cdot b =}{∣ a ∣∣ b ∣} = \frac{a _{1} b _{1} + a _{2} b _{2} + a _{3} b _{3}}{a _{1}^{2} + a _{2}^{2} + a _{3}^{2} \cdot b _{1}^{2} + b _{2}^{2} + b _{3}^{2}}$

Cross Product

The vector product of two nonzero vectors $a$ and $b$ , is denoted by $a \times b$ and defined as $a \times b = ∣ a ∥ b ∣ sin θ \overset{n}{^}$

Where, θ is the angle between $a$ and $b$ , 0 ≤ θ ≤ π and $\overset{n}{^}$ is a unit vector perpendicular to both $a$ and $b$ , such that $a$ , $b$ and $\hat{n}$ form a right handed system. i.e., the right handed system rotated from $a$ to $b$ moves in the direction of $\overset{n}{^}$ .

If either $a = 0 or b = 0$ , then θ is not defined and in this case , we define $a \times b = 0$ .

2.0Vector Algebra Formulas

Vector Addition

$A + B = (A_{x} + B_{x}) \hat{i} + (A_{y} + B_{y}) \hat{j} + (A_{z} + B_{z}) \hat{k}$

Combine corresponding components of vectors A and B.

Vector Subtraction

$A - B = (A_{x} - B_{x}) \hat{i} + (A_{y} - B_{y}) \hat{j} + (A_{z} - B_{z}) \hat{k}$

Subtract corresponding components of vector B and A.

Scalar Multiplication

$k A = (k A_{x}) \hat{i} + (k A_{y}) \hat{j} + (k A_{z}) \hat{k}$

Multiply each component of vector A by the scalar k.

Dot Product

$A \cdot B = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}$

Calculate the sum of the products of the corresponding components of vectors A and B.

Alternatively, using magnitudes and the cosine of the angle θ between the vectors:

$A \cdot B = ∣ A ∥ B ∣ cos θ$

Note: If $A$ and $B$ are Parallel then $A \times B$ are Perpendicular then $A \cdot B = 0$

Cross Product

$A \times B = (A_{y} B_{z} - A_{z} B_{y}) \hat{i} - (A_{x} B_{z} - A_{z} B_{x}) \hat{j} + (A_{x} B_{y} - A_{y} B_{x}) \hat{k}$

Compute a vector perpendicular to both A and B with magnitude equal to the area of the parallelogram they span.

Vector Magnitude

$∣ A ∣ = A_{x}^{2} + A_{y}^{2} + A_{z}^{2}$

Calculate the length of vector A using its components.

Unit Vector

$A = \frac{A}{∣ A ∣}$

Divide vector A by its magnitude to get a unit vector in the same direction.

Projection of A onto B

$proj_{B} A = (\frac{A \cdot B}{∣ B ∣ ^{2}}) B$

Project vector A onto B, resulting in a vector parallel to B.

Angle Between Two Vectors

$cos θ = \frac{A \cdot B}{∣ A ∥ B ∣}$

Find the cosine of the angle between vectors A and B.

3.0Vector Algebra Examples

Example 1: Find unit vector of $A = 3 \hat{i} + 4 \hat{j} - 5 \hat{k}$

Solution:

$∣ A ∣ = 9 + 16 + 25 = 50$ ⇒ unit vector: $\frac{3 i ^ + 4 j ^ - 5 k ^}{50}$

Example 2: If $A \equiv (2 \hat{i} + 3 \hat{j}), B \equiv (p \hat{i} + 9 \hat{j})$ and $C \equiv (\hat{i} - \hat{j})$ are collinear, then the value of p is:

(A) 1/2 (B) 3/2 (C) 7/2 (D) 5/2

Ans. (C)

Solution:

$AB = (p - 2) \hat{i} + 6 \hat{j}, AC = - \hat{i} - 4 \hat{j}$

Now A, B, C are collinear ⇔ $A B ∥ A C$ ⇔ $\frac{p - 2}{- 1} = \frac{6}{- 4}$ ⇔ $p = \frac{7}{2}$

Example 3: The value of λ when $a = 2 \hat{i} - 3 \hat{j} + \hat{k}$ and $b = 8 \hat{i} + λ \hat{j} + 4 \hat{k}$ are parallel is:

(A) 4 (B) –6 (C) –12 (D) 1

Ans. (C)

Solution:

Since $a$ & $b$ are parallel ⇒ $\frac{2}{8}$ = $- \frac{3}{λ}$ = $\frac{1}{4}$ ⇒ –12

Example 4: If $a$ and $b$ are unit vectors, then which of the following values of $a \cdot b$ is not possible?

(A) $3$ (B) $\frac{3}{2}$

(C) $\frac{1}{2}$ (D) $\frac{- 1}{2}$

Ans. (A)

Solution:

Since $a \cdot b = ∣ a ∣∣ b ∣ cos θ = cos θ$ and cosθ never be equal to $3$

Example 5: If p^th, q^th, r^th terms of a G.P. are the positive numbers a, b, c then angle between the vectors $lo g a^{2} \hat{i} + lo g b^{2} \hat{j} + lo g c^{2} \hat{k}$ and (q-r) $\hat{i} + (r - p) \hat{j} + (p - q) \hat{k}$ is:

(A) $\frac{p}{3}$

(B) $\frac{p}{2}$

(C) $sin^{- 1} (\frac{1}{a ^{2} + b ^{2} + c ^{2}})$

(D) none of these

Ans. (B)

Solution:

Let x₀ be first term and x the common ratio of the G.P.

∴ a = x₀ x^p-1, b = x₀x^q–1, c = x₀x^r-1

⇒ log a = log x₀ + (p–1) log x ; log b = log x₀ +(q–1) log x; log c = log x₀ +(r–1) log x

If $a = lo g a^{2} \hat{i} + lo g b^{2} \hat{j} + lo g c^{2} \hat{k}$ and $b = (q - r) \hat{i} + (r - p) \hat{j} + (p - q) \hat{k}$

$∴ a \cdot b = Σ2 (lo g a) (q - r) = 2 \sum (lo g x_{0} + (p - 1) lo g x) (q - r) = 0$

⇒ $a^{\land} b = \frac{π}{2} .$

Example 6: A unit vector perpendicular to the plane determined by the points (1,–1,2), (2,0,–1) and (0,2,1) is

(A) $\pm \frac{1}{6} (2 \hat{i} + \hat{j} + \hat{k})$

(B) $\frac{1}{6} (2 \hat{i} + \hat{j} + \hat{k})$

(C) $\frac{1}{6} (\hat{i} + \hat{j} + \hat{k})$

(D) $\frac{1}{6} (2 \hat{i} - \hat{j} - \hat{k})$

Ans. (A)

Solution:

$a = \hat{i} + \hat{j} - 3 \hat{k}, b = - 2 \hat{i} + 2 \hat{j} + 2 \hat{k}$

$a \times b = \hat{i} 1 - 2 \hat{j} 12 \hat{k} - 3 2 = 8 \hat{i} + 4 \hat{j} + 4 \hat{k}$

Hence unit vector = $\pm \frac{2 i ^ + j ^ + k ^}{6}$

Example 7: If $a = 2 \hat{i} + 3 \hat{j} - 5 \hat{k}, b = m \hat{i} + n \hat{j} + 12 \hat{k}$ and $a \times b = 0$ then (m,n)=

(A) $(- \frac{24}{5}, \frac{36}{5})$ (B) $(\frac{24}{5}, - \frac{36}{5})$

(C) $(- \frac{24}{5}, - \frac{36}{5})$ (D) $(\frac{24}{5}, \frac{36}{5})$

Ans. (C)

Solution:

$a \times b = \hat{i} 2 m \hat{j} 3 n \hat{k} - 5 12$ = $(36 + 5 n) \hat{i} - (24 + 5 m) \hat{j} + (2 n - 3 m) \hat{k}$ =0

$\Rightarrow m = \frac{- 24}{5}, n = \frac{- 36}{5}$

4.0Solved Questions on Vector Algebra

What is a unit vector?

Ans: A unit vector is a vector with a magnitude of one. It is used to indicate direction. If A is any vector, its unit vector is $A = \frac{A}{∣ A ∣}$ , where $∣ A ∣$ is the magnitude of A.

How do you find the magnitude of a vector?

Ans: The magnitude of a vector $A = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}$ is calculated as $∣ A ∣ = A_{x}^{2} + A_{y}^{2} + A_{z}^{2}$

What is the projection of one vector onto another?

Ans: The projection of vector A onto vector B is given by $proj_{B} A = (\frac{A \cdot B}{∣ B ∣ ^{2}}) B$ . It represents the component of A in the direction of B.

What is a position vector?

Ans: A position vector is a vector that represents the position of a point in space relative to an origin. For a point P(x, y, z), the position vector is $r = x \hat{i} + y \hat{j} + z \hat{k}$ .

How do you subtract one vector from another?

Ans: To subtract vector B from vector A, you subtract their corresponding components:

$A - B = (A_{x} - B_{x}) \hat{i} + (A_{y} - B_{y}) \hat{j} + (A_{z} - B_{z}) \hat{k}$

Also Read:

Coplanar Vectors	Vector Triple Product	Components of a Vector
Scalar Triple Product	Vector Algebra Previous Year Questions with Solutions	Multiplication of a Vector
Vectors in Maths	Collinear Vectors	Angle Between Two Vectors

1.0Learning Outcomes

After completing this chapter, you will be able to:

Understand the concept of vectors and scalars.
Perform vector addition and subtraction.
Represent vectors in component form.
Calculate the magnitude and direction of vectors.
Apply scalar and vector products.
Find the angle and projection between vectors.
Solve JEE Main, JEE Advanced, and CBSE Board questions confidently.

2.05-Minute Quick Revision

Must-Revise Concepts

Scalars and Vectors
Position Vector
Magnitude of a Vector
Unit Vector
Addition and Subtraction of Vectors
Direction Cosines
Section Formula
Scalar (Dot) Product
Vector (Cross) Product
Projection of a Vector

Important Formula Recall

Magnitude: $∣ a ∣ = a_{x}^{2} + a_{y}^{2} + a_{z}^{2}$
Unit Vector: $\overset{a}{^} = \frac{a}{∣ a ∣}$
Dot Product: $a \cdot b = ∣ a ∣∣ b ∣ cos θ$
Cross Product: $∣ a \times b ∣ = ∣ a ∣∣ b ∣ sin θ$

Important Properties / Shortcut Tips

Use the dot product to determine the angle between vectors.
The dot product is zero for perpendicular vectors.
The cross product is zero for parallel vectors.
Always simplify vectors into component form before calculations.
Normalize vectors before finding projections.

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Vector Operations	High	Easy–Moderate
Position & Unit Vectors	High	Easy
Scalar (Dot) Product	Very High	Moderate
Vector (Cross) Product	Very High	Moderate
Projection of Vectors	High	Moderate
Angle Between Vectors	Very High	Moderate
Mixed Vector Problems	Very High	High

4.0Formula Handbook

Available soon

5.0Common Mistakes & JEE Tips

Common Mistake	Correct Approach
Confusing dot product with cross product	Use the dot product for angles and the cross product for perpendicular vectors and area.
Incorrect vector components	Express vectors in component form before calculations.
Forgetting unit vectors	Normalize vectors while finding direction or projection.
Sign errors during vector addition	Add components carefully along each axis.
Wrong angle formula	Apply the dot product formula while calculating the angle between vectors.
Ignoring vector direction	Consider both magnitude and direction in every operation.
Algebraic mistakes	Simplify expressions before performing vector operations.

6.0ALLEN Faculty Tips

Practice vector operations in component form regularly.
Memorize the properties of dot and cross products.
Learn the geometric interpretation of vector products.
Revise projection and angle formulas frequently.
Strengthen vector concepts before studying Three-Dimensional Geometry.
Solve Previous Years' Questions (PYQs) for faster problem-solving.

7.0PYQ Trend Analysis

Vector Algebra is one of the most scoring chapters in JEE Main, JEE Advanced, and CBSE Board examinations. Questions are commonly asked on vector operations, dot product, cross product, projection, and geometric applications.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Vector Operations	1	Easy
Dot Product	1–2	Moderate
Cross Product	1–2	Moderate
Projection & Direction	1	Moderate
Mixed Vector Problems	1–2	Moderate–High

8.0Most Common Question Types

Finding the magnitude and direction of vectors.
Solving vector addition and subtraction problems.
Calculating the angle between two vectors.
Applying scalar and vector products.
Finding the projection of one vector on another.
Determining perpendicular or parallel vectors.
Solving mixed vector algebra problems.
Concept-based and previous years' examination questions.

9.0Smart Revision Strategy

Revise vector notation and component form.
Practice magnitude and unit vector problems.
Learn dot product applications.
Revise cross product properties.
Practice angle and projection questions.
Solve mixed vector algebra problems.
Practice Previous Years' Questions (PYQs).
Complete revision with a timed mock test.

Test your Knowledge

question 1 of 5

A quantity having both magnitude and direction is called a:

Frequently Asked Questions

What is the difference between a scalar and a vector?

What is the significance of the dot product in Vector Algebra?

When is the vector (cross) product used?

What is a unit vector and why is it important?

Why is Vector Algebra important for JEE Main and JEE Advanced?

Join ALLEN!

Class 12 Maths Chapter 10: Vector Algebra

1.0Chapter Snapshot

2.0Chapter 10 Smart Study Material by ALLEN Experts: Download PDFs

Test your Knowledge

question 1 of 5

A quantity having both magnitude and direction is called a:

Vector Algebra

1.0What is Vector Algebra

2.0Types of Vectors in Vector Algebra

Zero Vector (Null Vector)

Unit Vector

Position Vector

Co-initial Vectors

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties / Shortcut Tips

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

Frequently Asked Questions

What is the difference between a scalar and a vector?

What is the significance of the dot product in Vector Algebra?

When is the vector (cross) product used?

What is a unit vector and why is it important?

Why is Vector Algebra important for JEE Main and JEE Advanced?

Join ALLEN!

Class 12 Maths Chapter 10: Vector Algebra

1.0Chapter Snapshot

2.0Chapter 10 Smart Study Material by ALLEN Experts: Download PDFs

Vector Algebra

1.0What is Vector Algebra

2.0Types of Vectors in Vector Algebra

Zero Vector (Null Vector)

Unit Vector

Position Vector

Co-initial Vectors

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

Important Properties / Shortcut Tips

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy