✕ Cross Product Calculator
Cross Product Calculator
Compute a × b for two 3D vectors. Shows determinant expansion, i/j/k components, result vector, magnitude, and verifies a·(a×b) = 0.
Cross Product a × b
3D vectors only — i, j, k components
✓ Step-by-Step✓ Magnitude✓ Free⚡ Loading SymPy Engine…
Vector a = (a₁, a₂, a₃)
×
Vector b = (b₁, b₂, b₃)
Quick Examples
Cross Product Formula
The cross product of two 3D vectors a and b is a vector perpendicular to both:
$$\mathbf{a} \times \mathbf{b} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\end{vmatrix} = (a_2b_3-a_3b_2)\mathbf{i} - (a_1b_3-a_3b_1)\mathbf{j} + (a_1b_2-a_2b_1)\mathbf{k}$$
Key Properties
- a × b is perpendicular to both a and b
- |a × b| = |a||b|sin(θ) = area of parallelogram formed by a and b
- a × a = 0 (cross product with itself is zero)
- a × b = −(b × a) (anti-commutative)
- a · (a × b) = 0 (perpendicularity confirmed)
Right-Hand Rule
Point your right hand fingers from a toward b. Your thumb points in the direction of a × b.