The vector cross product calculator computes A×B, the dot product A·B, vector magnitudes, the angle between vectors, and unit vectors — with full determinant expansion steps for 3D vectors.
Vector Cross Product Calculator
A × B (Cross Product)
Step-by-Step
How to Use the Vector Cross Product Calculator
The cross product is a fundamental operation in 3D geometry and physics. It produces a vector perpendicular to both inputs with magnitude equal to the area of the parallelogram formed by A and B.
Cross Product Formula
For A = (a₁,a₂,a₃) and B = (b₁,b₂,b₃), the cross product A×B = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁). This is the cofactor expansion of the 3×3 determinant with unit vectors î, ĵ, k̂ in the first row.
Dot Product and Angle
The dot product A·B = a₁b₁ + a₂b₂ + a₃b₃ = |A||B|cos(θ). To find the angle: θ = arccos(A·B / (|A||B|)). If A·B = 0, the vectors are perpendicular. If A×B = (0,0,0), the vectors are parallel (or one is zero).
Example
A = (1,2,3), B = (4,5,6). Cross product: A×B = (2×6−3×5, 3×4−1×6, 1×5−2×4) = (12−15, 12−6, 5−8) = (−3, 6, −3). Dot product: 1×4+2×5+3×6 = 4+10+18 = 32. Verify: (−3)×4 + 6×5 + (−3)×6 = −12+30−18 = 0 ✓ (A×B is perpendicular to both A and B).
FAQ
What is the cross product?
The cross product A × B of two 3D vectors gives a third vector perpendicular to both A and B. Its magnitude equals |A||B|sin(θ), where θ is the angle between A and B — equal to the area of the parallelogram formed by A and B.
How is the cross product calculated?
For A = (a₁,a₂,a₃) and B = (b₁,b₂,b₃): A×B = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁). This is the determinant expansion of the 3×3 matrix with i,j,k unit vectors in the first row.
What is the difference between cross product and dot product?
The dot product A·B = a₁b₁+a₂b₂+a₃b₃ is a scalar equal to |A||B|cos(θ). It measures how parallel the vectors are. The cross product A×B is a vector measuring how perpendicular they are. If A·B = 0, vectors are perpendicular; if A×B = 0, they're parallel.
Is this calculator free?
Yes, completely free with no signup required. All calculations run in your browser.
What is the right-hand rule?
The right-hand rule determines the direction of A×B: point fingers toward A, curl toward B, and the thumb points in the direction of A×B. Note: B×A = −(A×B), so cross products are anti-commutative.