Cross Product Calculator – 3D Vectors
Calculate the cross product of two 3D vectors with step-by-step solution.
Cross Product: Definition and Applications
The cross product (also called vector product) of two 3D vectors A and B produces a third vector that is perpendicular to both. It is only defined in 3-dimensional (and 7-dimensional) space, unlike the dot product which works in any dimension.
| Vector A | Vector B | Cross Product A × B | Magnitude |
|---|---|---|---|
| (1, 0, 0) | (0, 1, 0) | (0, 0, 1) | 1 |
| (2, 3, 4) | (5, 6, 7) | (−3, 6, −3) | 7.35 |
| (1, 2, 3) | (4, 5, 6) | (−3, 6, −3) | 7.35 |
The cross product formula: A × B = (AyBz−AzBy, AzBx−AxBz, AxBy−AyBx). The magnitude equals |A||B|sin(θ), which equals the area of the parallelogram formed by the two vectors. Applications: computing surface normals in 3D graphics, torque in physics, and angular momentum in mechanics.
What is the difference between cross product and dot product?
Dot product: A · B = |A||B|cos(θ), scalar result, measures projection/alignment. Cross product: A × B, vector result perpendicular to both, measures area/rotation. Dot product is zero for perpendicular vectors; cross product is zero for parallel vectors.
Is the cross product commutative?
No — it is anti-commutative: A × B = −(B × A). The direction flips when you swap the operands.