Cross Product Calculator
Calculate the cross product of two 3D vectors — the resulting vector's components, magnitude, unit vector, parallelogram/triangle area, and the angle between the two input vectors.
Input
Output
| Metric | Value |
|---|---|
| No data yet | |
More ways to use this tool
REST API
curl -X POST https://api.iotools.cloud/v1/tool/cross-product-calculator \
-H "Authorization: Bearer YOUR_API_KEY" \
-H "Content-Type: application/json" \
-d '{
"ax": "1",
"ay": "0",
"az": "0",
"bx": "0",
"by": "1",
"bz": "0"
}'Swap in your own key from your account. The tool's fields are the body — no wrapper.
Ask an AI agent
Use the IOTools `cross-product-calculator` tool (Cross Product Calculator) on this input:
YOUR_INPUT_HEREPaste this at any agent connected to the IOTools MCP server, then add your input.
Embed widget
<iframe
src="https://iotools.cloud/embed/cross-product-calculator/"
width="100%" height="520" frameborder="0" scrolling="no" loading="lazy"
title="Cross Product Calculator — iotools.cloud"
sandbox="allow-scripts allow-forms allow-same-origin allow-downloads allow-popups allow-popups-to-escape-sandbox"
allow="clipboard-write"
style="width:100%;border:1px solid #e5e7eb;border-radius:12px;overflow:hidden"></iframe>
<script src="https://iotools.cloud/embed.js" async></script>Drop this into your own page — free, no key required, just a link back.
| Cost per API/MCP call | From 5 credits |
|---|---|
| Need more credits? | View pricing |
Also available with
Guides
What the cross product gives you
The cross product a × b of two 3D vectors is a third vector that's perpendicular to both a and b, with a magnitude equal to the area of the parallelogram they span. It shows up constantly in physics and 3D graphics: computing a surface normal, torque, angular momentum, or the area of a triangle defined by three points.
The formula used
For a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃):
a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)From there, the calculator also derives:
- Magnitude of a, b, and a × b (via √(x² + y² + z²))
- Unit cross product — a × b divided by its own magnitude, giving a direction with no length
- Parallelogram area — equal to |a × b|
- Triangle area — half the parallelogram area, useful for the area of a triangle with two sides a and b
- Dot product, used to get the angle between a and b via cos θ = (a · b) / (|a||b|)
- A perpendicularity check — a · (a × b) and b · (a × b), which are always 0 in exact arithmetic and confirm the cross product really is perpendicular to both inputs
The right-hand rule
The direction of a × b (not just its components) follows the right-hand rule: point your fingers along a, curl them toward b, and your thumb points along a × b. Swapping the order flips the sign — b × a = −(a × b) — which is why cross product order matters and, unlike the dot product, isn't commutative.
When is the cross product the zero vector?
Whenever a and b are parallel (including when one is the zero vector) — there's no unique perpendicular direction to point in, and the "area" they span collapses to zero. The calculator flags this explicitly rather than showing a division-by-zero unit vector.
How to use it
- Enter the x, y, z components of vector a.
- Enter the x, y, z components of vector b.
- Read the cross product vector, its magnitude/unit form, the areas it implies, and the angle between the two original vectors.
FAQ
Can I use this for 2D vectors? Yes — leave the z component at 0 for both vectors. The result's x and y components will also be 0, leaving only a z component, which is the standard "2D cross product" (a scalar) most people actually want.
Why is my cross product's z component nonzero when I only entered x, y values? Check that you actually left both z fields at 0 — if either vector has a nonzero z, the vectors aren't coplanar with the xy-plane and the cross product won't be purely vertical.
Does this calculator store my data? No. Everything runs in your browser — your vectors and results are never sent to or stored on our servers.