Cross Product Calculator (3D Vectors)

The cross product calculator takes two 3D vectors a and b and instantly returns every quantity you need: the cross product vector a × b, its magnitude, the dot product, the angle between the vectors, the parallelogram and triangle areas, all three unit vectors and the optional scalar triple product. It is built for physics students, engineering undergraduates, 3D graphics developers and anyone who needs a quick, reliable sanity-check of vector geometry.

How it works

Given a = (ax, ay, az) and b = (bx, by, bz), the tool evaluates:

• Cross product: a × b = (ay·bz − az·by, az·bx − ax·bz, ax·by − ay·bx) — a new vector perpendicular to both a and b, whose direction follows the right-hand rule.
• Magnitude: |a × b| = |a||b|sin(θ) — equals the area of the parallelogram spanned by a and b.
• Dot product: a · b = ax·bx + ay·by + az·bz = |a||b|cos(θ) — used together with the magnitude to recover the angle.
• Angle: θ = arccos((a · b) / (|a||b|)), in degrees, clamped to the valid −1 … 1 range so floating-point rounding never produces NaN.
• Unit vectors: â = a / |a|, b̂ = b / |b| and the unit normal n̂ = (a × b) / |a × b| — all length 1 in the direction of their parent vector.
• Triangle area: ½|a × b| — exactly half the parallelogram area.
• Scalar triple product: a · (b × c) — the signed volume of the parallelepiped formed by three vectors; zero means the three vectors are coplanar.

The formula box beneath the results shows each intermediate multiplication so you can follow the arithmetic step by step.

Worked example

Let a = (1, 2, 3) and 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)

Verification: |a||b|sin(θ) = 3.742 × 8.775 × sin(12.93°) ≈ 7.348 — matches |a × b| exactly. The dot product 1·4 + 2·5 + 3·6 = 4 + 10 + 18 = 32, and |a||b|cos(θ) = 3.742 × 8.775 × cos(12.93°) ≈ 32 — consistent.

Key properties of the cross product

The cross product is anti-commutative: swapping the inputs negates the result, so b × a = −(a × b). It is also distributive over addition: a × (b + c) = a × b + a × c. Unlike the dot product, the cross product is only defined in three (and seven) dimensions. In 2D, the “cross product” refers to just the z-component: ax·by − ay·bx, which is the signed area of the parallelogram and is what the 2D vector calculator uses.

The result vector is always orthogonal to both inputs. You can confirm this by computing (a × b) · a and (a × b) · b — both will be 0 (up to floating-point precision). This orthogonality property is why cross products are fundamental in 3D graphics (surface normals), physics (torque τ = r × F, magnetic force F = qv × B) and robotics (rotation axes).