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MathematicsClass 12Vector Algebra

Vector Algebra | Class 12 Mathematics Notes

By ConceptScroll Team · Published on 17 July 2026 · 3 min read

Vector Algebra – this guide gives you a concise, exam-ready overview of Vector Algebra from Class 12 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.

Introduction

Vector Algebra is a fundamental branch of mathematics that deals with vectors, which are quantities characterized by both magnitude and direction. Unlike scalars, which have only magnitude (such as mass, temperature, or speed), vectors provide more comprehensive information by incorporating direction. This makes vectors essential in describing physical quantities like displacement, velocity, acceleration, and force. The study of vectors involves understanding their representation, types, and algebraic operations such as addition, subtraction, and multiplication. Vector Algebra forms the basis for many applications in physics, engineering, and computer science, enabling the analysis and solution of problems involving directional quantities. This chapter introduces the concept of vectors, their types, and the algebraic operations that can be performed on them, laying the groundwork for advanced mathematical and physical studies.

📊 Diagram: Diagram illustrating a vector represented as an arrow with a specific length (magnitude) and direction, contrasted with a scalar represented by a simple numerical value.

🧪 Activity: No specific activity in this introductory section.

🔗 Connection: This section leads to the next section 'Types of Vectors' where different categories of vectors are defined and explained.

Frequently asked questions

Exercise 4.1 1. Represent graphically the following vectors in the Cartesian plane: (i) a = 2i + 3j (ii) b = -i + 4j (iii) c = 3i - 2j

To represent the vectors graphically: (i) a = 2i + 3j This vector starts from the origin (0,0) and ends at the point (2,3). Draw an arrow from (0,0) to (2,3). (ii) b = -i + 4j This vector starts from the origin and ends at (-1,4). Draw an arrow from (0,0) to (-1,4). (iii) c = 3i - 2j This vector starts from the origin and ends at (3,-2). Draw an arrow from (0,0) to (3,-2).

Exercise 4.1 2. Write each of the following vectors in terms of i, j, k: (i) The vector from A(1, 2, 3) to B(4, 6, 8) (ii) The vector from P(-2, 0, 5) to Q(3, -4, 7)

(i) The vector from A(1, 2, 3) to B(4, 6, 8) is given by: AB = (4 - 1)i + (6 - 2)j + (8 - 3)k = 3i + 4j + 5k (ii) The vector from P(-2, 0, 5) to Q(3, -4, 7) is: PQ = (3 - (-2))i + (-4 - 0)j + (7 - 5)k = 5i - 4j + 2k

Exercise 4.1 3. Find the magnitude of the following vectors: (i) a = 2i + 3j + 6k (ii) b = -i + 4j - 2k

(i) |a| = sqrt(2^2 + 3^2 + 6^2) = sqrt(4 + 9 + 36) = sqrt(49) = 7 (ii) |b| = sqrt((-1)^2 + 4^2 + (-2)^2) = sqrt(1 + 16 + 4) = sqrt(21)

Exercise 4.1 4. If a = 2i - j + k and b = i + 2j - 3k, find a + b and a - b.

a + b = (2i - j + k) + (i + 2j - 3k) = (2+1)i + (-1+2)j + (1-3)k = 3i + j - 2k a - b = (2i - j + k) - (i + 2j - 3k) = (2-1)i + (-1-2)j + (1-(-3))k = i - 3j + 4k

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