📐 CBSE · Class 12 · Mathematics · Chapter 10

Vector
Algebra

Complete chapter resources for CBSE Class 12 Maths — topic breakdown, key formulas, sample questions, previous year board questions, and instant AI question paper generation.

4Topics
6–8Board marks
8Sample questions
3PYQ included
Sign Up Free & Generate → See sample questions ↓

Free for independent teachers · No credit card required

Key Formulas — Chapter 10
  • Magnitude: |a⃗| = √(a₁² + a₂² + a₃²)
  • Unit vector: â = a⃗ / |a⃗|
  • Dot product: a⃗ · b⃗ = |a||b| cos θ
  • Cross product: |a⃗ × b⃗| = |a||b| sin θ
  • Projection of a⃗ on b⃗: (a⃗ · b⃗) / |b⃗|
  • l² + m² + n² = 1 (direction cosines)
Chapter Overview

What this chapter covers

Chapter 10 introduces the algebra of vectors — quantities that possess both magnitude and direction. The chapter begins by distinguishing scalars (e.g., mass, temperature) from vectors (e.g., displacement, force), then establishes the formal language of position vectors, direction cosines, and direction ratios needed to describe points and lines in three-dimensional space.

The two central operations are the scalar (dot) product — a⃗ · b⃗ = |a||b| cos θ, which yields a number and is used to find angles between vectors and compute projections — and the vector (cross) product — a⃗ × b⃗ = |a||b| sin θ n̂, which yields a vector perpendicular to both and is used to find areas of parallelograms and triangles. Both products are tested extensively in CBSE board papers, with the cross product also providing the backbone for Chapter 11 (Three Dimensional Geometry).

Board questions on this chapter range from straightforward magnitude and unit-vector calculations (1–2 marks) to multi-step problems asking for the angle between two vectors, projection along a direction, or area of a triangle given position vectors of its vertices (3–5 marks). Students are expected to know when vectors are perpendicular (dot product = 0), parallel (cross product = 0), and how to express any vector as a linear combination of standard basis vectors i̊, j̊, k̊.

Topics

What's inside Chapter 10

As per NCERT Class 12 Mathematics (CBSE syllabus)

Topic 1
Scalars, Vectors & Types of Vectors
Definitions of scalar and vector quantities. Types: zero vector, unit vector, co-initial, collinear, equal, and negative vectors. Position vectors and notation in component form (a₁î + a₂ĵ + a₃k̂).
Topic 2
Addition, Subtraction & Section Formula
Triangle and parallelogram laws of vector addition. Subtraction as addition of the negative vector. Section formula: position vector of a point dividing a line segment in ratio m:n internally and externally.
Topic 3
Scalar (Dot) Product
Definition a⃗ · b⃗ = |a||b| cos θ, properties (commutative, distributive). Applications: angle between two vectors, perpendicularity condition (a⃗ · b⃗ = 0), projection of one vector on another.
Topic 4
Vector (Cross) Product
Definition a⃗ × b⃗ = |a||b| sin θ n̂, computation using the 3×3 determinant form. Properties (anti-commutative, distributive). Applications: area of parallelogram/triangle, unit vector perpendicular to two vectors.
Connections

How this chapter fits in

Useful for setting question difficulty and cross-chapter papers.

Builds on
Class 11 · Introduction to 3D Geometry
Coordinates in space, distance formula, section formula in 3D
Class 11 · Trigonometry
cos θ and sin θ identities used in dot and cross product definitions
Chapter 10 Vector
Algebra
Leads to
Ch 11 · Three Dimensional Geometry
Vector equations of lines and planes, angle between planes
Engineering & Physics (JEE / NEET)
Work, torque, flux — all rely on dot and cross products
Board Blueprint

Marks & question-type breakdown

Typical pattern based on CBSE Class 12 board papers from the last five years.

Question type Marks Typical count What's usually tested
MCQ / Assertion-Reason 1 1–2 Magnitude, unit vector, perpendicularity or parallelism condition
Very Short Answer 2 1 Find angle between two vectors, unit vector, or projection
Short Answer 3 1 Dot/cross product application, area of triangle/parallelogram
Long Answer 5 0–1 Multi-step problem combining section formula with dot/cross product
Total (approximate) 6–8 3–5 Weightage varies across paper sets and years
Sample Questions

8 sample questions — generated by MarksZen AI

Aligned to CBSE Class 12 Maths Chapter 10. Covers all question types across Easy, Medium, and Hard difficulty.

Q1 Easy 1 mark MCQ
If a⃗ = 2î − ĵ + 2k̂, the magnitude |a⃗| is: (a) 3 (b) √5 (c) 4 (d) √7
Q2 Easy 2 marks Short Answer
Find the unit vector in the direction of a⃗ = 3î + 4ĵ − 0k̂.
Q3 Medium 2 marks Short Answer
Find the angle between vectors a⃗ = î + ĵ − k̂ and b⃗ = î − ĵ + k̂.
Q4 Medium 2 marks Short Answer
If a⃗ = 2î + 3ĵ + k̂ and b⃗ = î − ĵ + 2k̂, find the projection of a⃗ on b⃗.
Q5 Medium 3 marks Short Answer
Find the area of the parallelogram whose adjacent sides are given by the vectors a⃗ = 3î + ĵ + 4k̂ and b⃗ = î − ĵ + k̂.
Q6 Hard 4 marks Word Problem
The position vectors of three points A, B, and C are 2î + 4ĵ − k̂, 4î + 5ĵ + k̂, and 3î + 6ĵ − 3k̂ respectively. (i) Find vectors AB⃗ and AC⃗. (ii) Hence, find the area of triangle ABC.
Q7 Hard 4 marks Word Problem
If a⃗ + b⃗ + c⃗ = 0⃗ and |a⃗| = 3, |b⃗| = 5, |c⃗| = 7, find the angle between a⃗ and b⃗.
Q8 Hard 5 marks Case-Based
A force F⃗ = 5î + 3ĵ + 2k̂ acts on a particle whose displacement is d⃗ = 2î − ĵ + k̂. (i) Find the work done by the force. (ii) Find a unit vector perpendicular to both F⃗ and d⃗. (iii) Verify that this unit vector is perpendicular to F⃗ using the dot product.
Generate a full paper with answer key →

MarksZen AI creates a complete question paper with answer key in under 2 minutes.

Previous Year Questions

From CBSE board examinations

Actual questions from past Class 12 Maths board papers — Vector Algebra chapter.

Board 20222 marks
If a⃗ · b⃗ = 0 and a⃗ × b⃗ = 0⃗, then what can you conclude about vectors a⃗ and b⃗? (All India 2022)
Board 20233 marks
Find a unit vector perpendicular to both the vectors a⃗ = 4î − ĵ + 3k̂ and b⃗ = 2î + ĵ − 2k̂. (Delhi 2023)
Board 20204 marks
Using vectors, find the area of the triangle ABC with vertices A(1, 2, 3), B(2, −1, 4), and C(4, 5, −1). (CBSE 2020)
For Teachers

Create a board-aligned
question paper in 2 minutes.

Pick chapter, set the question-type mix and total marks — MarksZen AI generates the full paper with answer key. CBSE, ICSE, and all State Boards supported.

  • All 4 topics of this chapter
  • MCQ + short answer + word problems
  • Answer key included
  • PDF export ready
Sign Up Free & Generate →
FAQ

Questions teachers ask

How many marks does Vector Algebra carry in the CBSE Class 12 board exam? +
Vector Algebra typically carries 6–8 marks in the CBSE Class 12 Mathematics board exam. Questions appear as one 2-mark short answer (usually a dot or cross product), one 3-mark question involving unit vectors or projections, and occasionally a 5-mark question combining vectors with 3D geometry or area of a parallelogram.
What is the difference between dot product and cross product, and which appears more in board exams? +
The dot (scalar) product a⃗ · b⃗ = |a||b| cos θ gives a scalar and is used to find angles between vectors and check perpendicularity (a⃗ · b⃗ = 0). The cross (vector) product a⃗ × b⃗ = |a||b| sin θ n̂ gives a vector and is used to find area of parallelograms/triangles and check collinearity. Both appear in board papers; dot product questions are slightly more frequent at the 2-mark level, while cross product is common in 3–5 mark problems involving area.
How do you find the unit vector along a given vector, and is it a common board question? +
The unit vector along a⃗ is â = a⃗ / |a⃗|, where |a⃗| = √(a₁² + a₂² + a₃²). Yes, finding a unit vector is a standard 1–2 mark question in CBSE Class 12 board papers and also appears as a sub-part in longer questions. Students should be comfortable computing magnitude and dividing each component by it.
What are direction cosines and direction ratios, and how are they tested? +
Direction ratios (a, b, c) are any numbers proportional to the direction cosines. Direction cosines (l, m, n) are the cosines of the angles a vector makes with the x, y, and z axes, satisfying l² + m² + n² = 1. Board questions ask students to find direction cosines of a given vector or verify that l² + m² + n² = 1 for a set of numbers — typically 2-mark questions that appear frequently.
How do I generate a custom question paper for Vector Algebra using MarksZen? +
Sign up for a free MarksZen account, choose CBSE Class 12 Mathematics, select Chapter 10 (Vector Algebra), set your preferred question-type mix (MCQ, short answer, word problem) and total marks — the AI generates a complete board-aligned paper with answer key in under 2 minutes, ready for PDF export.