Menu

Vector Algebra

NCERT Class 12 Mathematics Chapter 10

Vector Algebra for Class 12 Maths and JEE: types of vectors, addition, components, section formula, projections, and scalar and vector products.

5 topics Updated 8 Nov 2026

For: Class 12 and JEE

In the syllabus: JEE Class 12 syllabus

With your textbook

NCERT Mathematics Part II (Class 12), Chapter 10 “Vector Algebra”, reprint 2026-27.

Open the official NCERT chapter (PDF, 5.0 MB, ncert.nic.in)

Checked on 19 Oct 2026.

Learn in this order

  1. Vectors: basic concepts, position vector and direction cosines

    Direction cosines of a vector for Class 12 Maths and JEE: scalars and vectors, magnitude, position vector, l + m + n = 1, solved examples.

    NCERT Sections 10.1–10.2 10 min read 4 solved examples 10 practice questions

  2. Types of vectors: zero, unit, collinear, equal and negative

    Types of vectors for Class 12 Maths and JEE: zero and unit vectors, coinitial, collinear, equal and negative vectors, free vectors, with NCERT questions.

    NCERT Section 10.3 9 min read 3 solved examples 10 practice questions

  3. Addition of vectors: triangle and parallelogram laws

    Addition of vectors for Class 12 Maths and JEE: triangle law, parallelogram law, subtraction, commutative and associative properties, with NCERT questions.

    NCERT Section 10.4 10 min read 5 solved examples 10 practice questions

  4. Multiplying a vector by a scalar, components and the section formula

    Section formula in vectors for Class 12 Maths and JEE: internal, external and midpoint forms, scalar multiples, unit vectors, components, collinearity.

    NCERT Section 10.5 13 min read 7 solved examples 12 practice questions

  5. Scalar and vector products of two vectors

    Dot and cross products for Class 12 Maths and JEE: angle between vectors, projection, Cauchy–Schwarz, cross product as a determinant, areas.

    NCERT Section 10.6 15 min read 7 solved examples 12 practice questions

Key formulas in this chapter

Vectors: basic concepts

  • |r| = √(x + y + z) for r = OP, P(x, y, z)
  • l = x/r, m = y/r, n = z/r, r > 0
  • l + m + n = 1

Types of vectors

  • |0| = 0; |â| = 1
  • BA = −AB
  • equal vectors: same magnitude and same direction

Addition of vectors

  • AB + BC = AC; AB + BC + CA = 0
  • a + b = b + a; (a + b) + c = a + (b + c)
  • a + 0 = a; a + (−a) = 0

Scalar multiple and components

  • |λa| = |λ| |a|; â = a/|a| for a ≠ 0
  • r = xî + yĵ + zk̂, |r| = √(x + y + z)
  • PP = (x − x)î + (y − y)ĵ + (z − z)k̂
  • r = (mb + na)/(m + n), m,n > 0; external (mb − na)/(m − n), m ≠ n

Dot and cross products

  • a · b = |a| |b| cos θ = ab + ab + ab
  • projection of a on b = a · b / |b|, b ≠ 0
  • |a × b| = |a| |b| sin θ; b × a = −(a × b)
  • î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ
  • area of parallelogram = |a × b|; of triangle = ½|a × b|

More in Mathematics