CBSE Class 12 Maths: Vector Algebra — Complete Notes 2026

Types of Vectors

• Zero vector: magnitude 0; Unit vector: magnitude 1; â = a/|a|
• Collinear vectors: parallel or anti-parallel; same or opposite direction
• Equal vectors: same magnitude AND direction; position vector: from origin O to point P

Vector Operations

• Addition: triangle law or parallelogram law; commutative: a + b = b + a; associative: (a + b) + c = a + (b + c)
• Section formula: position vector of point dividing PQ in ratio m:n: r = (m×q + n×p)/(m+n)
• Component form: a = aᵢ î + aⱼ ĵ + aₖ k̂ where î, ĵ, k̂ are unit vectors along x, y, z axes

Dot (Scalar) Product

• a⃗ · b⃗ = |a||b| cos θ; also a⃗ · b⃗ = a₁b₁ + a₂b₂ + a₃b₃
• a⃗ · b⃗ = 0 ⟺ vectors perpendicular (if non-zero); a⃗ · a⃗ = |a|²
• Projection of a on b: (a⃗ · b⃗)/|b|; projection vector: [(a⃗ · b⃗)/|b|²] b⃗

Cross (Vector) Product

• a⃗ × b⃗ = |a||b| sin θ n̂; direction by right-hand rule; magnitude = area of parallelogram
• In component form: a⃗ × b⃗ = determinant with rows î,ĵ,k̂; a₁,a₂,a₃; b₁,b₂,b₃
• a⃗ × b⃗ = 0 ⟺ vectors parallel or anti-parallel; cross product is anti-commutative: a×b = −b×a

Applications of Vectors

• Area of parallelogram: |a⃗ × b⃗|; area of triangle: ½|a⃗ × b⃗|
• Scalar triple product: a⃗ · (b⃗ × c⃗) = |det| = volume of parallelepiped; zero ⟺ coplanar
• Moment of force: r⃗ × F⃗ (torque); work done: F⃗ · d⃗

Coplanarity and Collinearity

• Three points A, B, C are collinear if AB⃗ = λ AC⃗ for some scalar λ
• Four points coplanar if scalar triple product of any three edge vectors = 0
• Vector equation of line: r⃗ = a⃗ + λb⃗; passes through a⃗ in direction b⃗

CBSE Board Focus

• Vectors: 5–7 marks; dot product and cross product numericals, projection, angle between vectors
• Collinearity proof using vectors: show AB = kAC
• Scalar triple product: find volume of parallelepiped; check coplanarity — common 3-mark question