3D vector cross product and parallelogram area

How the estimate works

Cx = uy·vz − uz·vy; Cy = uz·vx − ux·vz; Cz = ux·vy − uy·vx. Magnitude and parallelogram area are √(Cx² + Cy² + Cz²). Each component needs its own four relevant inputs, even beside a zero multiplier. Area and zero/nonzero status need all six inputs. An invalid irrelevant component does not hide an independently known result. Exact decimal fractions of accepted numerical values are used before final rounding; rounded cross components are never reused for area or status. Reversing u and v negates the product and preserves the area.

Only two declared real vectors in three-dimensional Euclidean space with the stated axes and units. For area, interpret them as adjacent side displacements from a common origin. The product follows right-hand order and is not normalized. This does not infer vectors from point coordinates, calculate a unit normal, angle, dot product, projection, triangle area, higher-dimensional product or geographic result. No graph, unit conversion, torque or other physical model, measurement validation, uncertainty estimate or fit/safety guarantee. Exact zero/nonzero comparisons use accepted values without a tolerance; they do not prove properties of measured objects.

Check the filled example

Our chosen example: u = (2, −1, 2), v = (1, 2, 0). Then u × v = (−4, 2, 5); magnitude and parallelogram area are √(16 + 4 + 25) = √45 ≈ 6.7082 square coordinate units. These are mathematical example values, not measured or recommended dimensions.

Accepted input values

Vector u: x component

2

Vector u: y component

-1

Vector u: z component

2

Vector v: x component

1

Vector v: y component

2

Vector v: z component

0

Plan the next step

Use the standard calculator for separate arithmetic. Check vector order, the right-handed axis convention and consistent units before interpreting the result.