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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.

Scalar (Dot) Product of Vectors

The scalar product, also known as the dot product, is an operation that takes two vectors and returns a scalar quantity. For two vectors a and b, the scalar product is defined as a · b = |a||b|cosθ, where |a| and |b| are the magnitudes of vectors a and b, and θ is the angle between them. The dot product measures how much one vector extends in the direction of another. Algebraically, if a = (a1, a2, a3) and b = (b1, b2, b3), then their dot product is a · b = a1b1 + a2b2 + a3b3. The dot product is commutative, meaning a · b = b · a, and distributive over vector addition. It is zero if the vectors are perpendicular (θ = 90°), indicating orthogonality. The scalar product is widely used in physics to calculate work done, projection of vectors, and in computer graphics for lighting calculations.

📊 Diagram: Diagram showing two vectors a and b with angle θ between them, illustrating the projection of one vector on another and the geometric meaning of the dot product.

🧪 Activity: Activity: Calculate the dot product of given vectors and verify if they are perpendicular.

🔗 Connection: This section leads to 'Vector (Cross) Product of Vectors' where multiplication of vectors resulting in another vector is introduced.

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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