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Calculation Machine

Cross Product Calculator

Vector cross product in 3D.

Reviewed by the Calculation Machine team · How we check our maths

Your values

ⓘ Live results

Results
a × b
(-16, 11, 14)
Magnitude
23.93742
Angle between vectors
57.927°
Parallelogram area
23.93742
Triangle area
11.96871

Quick answer

The cross product produces a vector perpendicular to both inputs, with magnitude equal to the parallelogram area they span.

The formula

a × b = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁)

Worked example

Worked example

Using the values the calculator loads with.

Inputs

  • Vector a (x, y, z)3, -2, 5
  • Vector b (x, y, z)1, 4, -2

Result

  • a × b(-16, 11, 14)
  • Magnitude23.93742
  • Angle between vectors57.927°
  • Parallelogram area23.93742
  • Triangle area11.96871

How to use the Cross Product Calculator

  1. 1Enter the three components of vector a.
  2. 2Enter the three components of vector b.
  3. 3Read the resulting perpendicular vector.

Frequently asked questions

Is cross product commutative?

No — a × b = −(b × a).

What if vectors are parallel?

The cross product is the zero vector.

About this calculator