Vectors

Concept overview

A vector has both magnitude and direction. In 3D, a vector is a triple (a, b, c) with magnitude √(a² + b² + c²). Basic operations: addition (component-wise), scalar multiplication (component-wise), and two multiplication-like operations — dot product (scalar output) and cross product (vector output).

Dot product a⃗ · b⃗ = a₁b₁ + a₂b₂ + a₃b₃ = |a⃗||b⃗| cos θ. Zero iff vectors are perpendicular. Positive iff acute angle between them; negative iff obtuse.

Cross product a⃗ × b⃗ = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁) = |a⃗||b⃗| sin θ n̂, where n̂ is a unit vector perpendicular to both a⃗ and b⃗ given by the right-hand rule. Magnitude |a⃗ × b⃗| equals the area of the parallelogram spanned by a⃗ and b⃗.

Scalar triple product [a⃗ b⃗ c⃗] = a⃗ · (b⃗ × c⃗) equals the signed volume of the parallelepiped spanned by the three vectors. Zero iff the three are coplanar. Vector triple product a⃗ × (b⃗ × c⃗) = (a⃗ · c⃗) b⃗ − (a⃗ · b⃗) c⃗ — the BAC-CAB rule.

Key formulas & concepts

• Dot product: a⃗ · b⃗ = |a⃗||b⃗| cos θ
• Cross product magnitude: |a⃗ × b⃗| = |a⃗||b⃗| sin θ
• Perpendicular test: a⃗ · b⃗ = 0
• Parallel test: a⃗ × b⃗ = 0⃗
• Scalar triple product: [a⃗ b⃗ c⃗] = a⃗ · (b⃗ × c⃗); volume of parallelepiped
• Coplanarity: [a⃗ b⃗ c⃗] = 0
• Vector triple (BAC-CAB): a⃗ × (b⃗ × c⃗) = (a⃗ · c⃗) b⃗ − (a⃗ · b⃗) c⃗

Common mistakes to avoid

• Assuming cross product commutes — a⃗ × b⃗ = − b⃗ × a⃗.
• Confusing scalar triple with vector triple — one gives a number, the other a vector.
• Using |a⃗ · b⃗| when |a⃗||b⃗| cos θ has known sign — dot product can be negative.
• Applying the right-hand rule with the wrong orientation for the cross-product direction.

JEE Main relevance

JEE Main asks 1–2 vector problems every paper — usually one dot/cross product and one scalar-triple or coplanarity question.

JEE Advanced relevance

Advanced problems combine vectors with 3D geometry (find the vector equation of a plane) or use identities like a⃗ × (b⃗ × c⃗) creatively in geometric proofs.

Previous-year questions (PYQ)

Scalar triple product and vector-triple-product problems are common. Drill both.