Cross Product VisualiserJEE Main

a × b — the perpendicular with an area — interactive Mathematics simulation for IIT-JEE.

Concept

The cross product a × b is a vector perpendicular to both a and b (right-hand rule), with magnitude |a||b|sinθ — the area of the parallelogram they span. It's computed as a 3×3 determinant with î, ĵ, k̂ in the top row. Zero cross product ⟺ parallel vectors; it's anti-commutative: b × a = −(a × b).

Key formula

\vec a \times \vec b = \begin{vmatrix} \hat i & \hat j & \hat k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}; \quad |\vec a \times \vec b| = ab\sin\theta

Derivation

Expand the determinant: î(a₂b₃ − a₃b₂) − ĵ(a₁b₃ − a₃b₁) + k̂(a₁b₂ − a₂b₁).

Magnitude: |a×b|² = |a|²|b|² − (a·b)² (Lagrange's identity) = a²b²(1 − cos²θ) = a²b²sin²θ. Direction follows the right-hand rule: curl fingers from a to b, thumb gives a × b.

Scenarios to explore

Cross Product Visualiser — a × b — perpendicular vector whose length is an area.

Real-world applications

Torque τ = r × F and angular momentum L = r × p.
Magnetic force F = qv × B.
Normal vectors to planes; area vectors in flux integrals.

JEE exam tips

Unit vector ⊥ to both: (a × b)/|a × b| — instant answer format.
Area of triangle ABC = ½|AB × AC| — the workhorse of 3D geometry areas.
î × ĵ = k̂ cyclic; any anticyclic order gives a minus.

Common mistakes

Forgetting the MINUS on the ĵ component when expanding the determinant.
Treating cross product as commutative — it flips sign.
Using cos θ instead of sin θ for the magnitude.

Exam traps to avoid

a × b = 0 with nonzero a, b means PARALLEL (θ = 0 or π), not perpendicular.
(a × b)² in JEE notation means |a × b|² — expand via Lagrange, not component grinding.