Cross Product Calculator
Calculate the cross product (A×B) of two 3D vectors, find the dot product, angle between vectors, unit normal, parallelogram area, and triple scalar product.
Cross Product A×B
—
Magnitude |A×B| —
Extended More scenarios, charts & detailed breakdown ▾
A×B
—
|A×B| —
Professional Full parameters & maximum detail ▾
Normal Vector & Area
Unit Normal Vector n̂ —
Parallelogram Area —
Triple Product & Relationship
Triple Scalar Product A·(B×C) —
Relationship (parallel/perp/neither) —
Projection
Projection of A onto B —
How to Use This Calculator
- Enter Vector A (x, y, z) and Vector B (x, y, z).
- Click Calculate to get the cross product vector and its magnitude.
- Use the Dot Product tab for A·B scalar result.
- Use the Both tab for cross product + dot product + angle between vectors.
- The Professional tab adds unit normal, parallelogram area, triple scalar product, and direction cosines.
Formula
A×B = (AyBz−AzBy, AzBx−AxBz, AxBy−AyBx) | |A×B| = |A||B|sin(θ)
A·B = AxBx + AyBy + AzBz = |A||B|cos(θ)
Example
A=(1,2,3), B=(4,5,6) → A×B = (−3, 6, −3), |A×B| ≈ 7.35, A·B = 32.
Frequently Asked Questions
- The cross product A×B of two 3D vectors produces a new vector perpendicular to both A and B. Its magnitude |A×B| = |A||B|sin(θ) equals the area of the parallelogram formed by A and B.
- The dot product A·B = |A||B|cos(θ) produces a scalar. The cross product A×B produces a vector. The dot product is zero when vectors are perpendicular; the cross product is the zero vector when they are parallel.
- A×B = (AyBz−AzBy, AzBx−AxBz, AxBy−AyBx). For A=(1,2,3) and B=(4,5,6): cross = (2×6−3×5, 3×4−1×6, 1×5−2×4) = (12−15, 12−6, 5−8) = (−3, 6, −3).
- The unit normal is the cross product divided by its magnitude: n̂ = (A×B)/|A×B|. It points perpendicularly to the plane containing A and B.
- A·(B×C) gives the scalar triple product, equal to the volume of the parallelepiped formed by A, B, C. If the result is 0, the three vectors are coplanar.