Cross Product Calculator
Vector cross product in 3D.
Enter your values
Enter the vector a (x, y, z) you are working with.
Enter the vector b (x, y, z) you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- 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.
a × b = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁)
At a glance
| What it does | Vector cross product in 3D. |
|---|---|
| Category | Math |
| Inputs needed | Vector a (x, y, z), Vector b (x, y, z) |
| Main output | a × b |
| Formula | a × b = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁) |
| Cost | Free — no sign-up, no download |
How to use the Cross Product Calculator
- 1Enter the three components of vector a.
- 2Enter the three components of vector b.
- 3Read the resulting perpendicular vector.
Inputs explained
- Vector a (x, y, z)
- Enter the vector a (x, y, z) you are working with.
- Vector b (x, y, z)
- Enter the vector b (x, y, z) you are working with.
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
Results
- a × b(-16, 11, 14)
- Magnitude23.93742
- Angle between vectors57.927°
- Parallelogram area23.93742
- Triangle area11.96871
Frequently asked questions
Is cross product commutative?
No — a × b = −(b × a).
What if vectors are parallel?
The cross product is the zero vector.
What do I need to enter into the Cross Product Calculator?
Just 2 values: vector a (x, y, z), vector b (x, y, z). Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Cross Product Calculator work out the answer?
The cross product produces a vector perpendicular to both inputs, with magnitude equal to the parallelogram area they span. It applies the formula a × b = (a₂b₃−a₃b₂, a₃b₁−a₁b₃, a₁b₂−a₂b₁) and shows the working so you can check each step by hand.
Is the Cross Product Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.