Cross product calculator
(2, 3, 4) × (5, 6, 7) is (−3, 6, −3), and the parallelogram they span has area √54: enter two vectors in space and get the perpendicular vector and that area. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Runs in your browser. Free, unlimited — your data never leaves this page.
This cross product calculator multiplies two three-dimensional vectors with the right-hand rule formula and returns both the resulting vector a times b and the area of the parallelogram those vectors span. The area equals the Euclidean magnitude of the cross product, so a single deterministic pass gives you the perpendicular direction, the signed orientation in space, and a non-negative scalar area useful for geometry, physics, and graphics. Classic checks include i cross j equals k, so the unit pair one zero zero and zero one zero yields zero zero one with area one, and the general sample two three four crossed with five six seven yields negative three, six, negative three with area square root of fifty-four. The engine uses only finite arithmetic with no network calls, no sampling, and no date-dependent shortcuts.
What the cross product means and what you get
The cross product of two vectors in three-dimensional space is a third vector that is perpendicular to both inputs and whose direction follows the right-hand rule: point your fingers along a, curl them toward b, and your thumb shows the sense of a times b.
Algebraically, if a equals ax ay az and b equals bx by bz, the components are ay times bz minus az times by for the first entry, az times bx minus ax times bz for the second, and ax times by minus ay times bx for the third. That construction is antisymmetric, so b times a equals the negative of a times b, and it is bilinear in the two arguments.
The length of the result equals the area of the parallelogram spanned by a and b, which is also absolute value of a times b times sine of the included angle.
Where cross products fit in study, physics, and software
Students meet the cross product when a textbook moves from planar vectors to three-dimensional geometry, when a contest problem asks for the area of a parallelogram or triangle in space without calculus, or when they build a normal for a plane given two edge directions. Teachers can pre-generate answer keys for modest integer components without trusting opaque calculator apps that silently drop a minus sign on the middle component.
Developers call the same contract from scripts to seed deterministic test vectors for mesh normals, to check torque directions in simple rigid-body demos, or to satisfy programming-challenge statements that demand both a times b and the parallelogram area rather than a hand-waved sketch. Contest practice often needs a fast, offline source of oriented products so solutions can be checked without graphing calculators under time pressure.
Product and education tools use the rewrite when a learner types two triples and the UI must show a perpendicular result and a clean area value.
What you can do with it
Homework and geometry checks
Compute a×b and the parallelogram area for textbook 3D vector problems before graphing or finding surface normals.
Physics and graphics pipelines
Normalize torque directions, face normals, and parallelogram areas from user-entered triples in demos and labs.
FAQ
What does the cross product calculator cost?
It is free in your browser on this page.
How is a×b defined?
For a=(ax,ay,az) and b=(bx,by,bz), a×b = (ay·bz−az·by, az·bx−ax·bz, ax·by−ay·bx). The parallelogram area is the Euclidean magnitude of that result.
What happens if a vector is not three-dimensional?
Both a and b must be arrays of exactly three finite numbers; other lengths are not the classical 3D cross product.
Does this tool compute the dot product?
No. It only returns the vector cross product and the parallelogram area (||a×b||). Use a scalar-product tool if you need a·b.
Why are results rounded to a precision?
Default precision is 10 decimal places; you can set precision from 0 to 15.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/algebra/cross-product \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":[1,0,0],"b":[0,1,0]}'const res = await fetch("https://api.kit.forhosting.com/algebra/cross-product", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": [
1,
0,
0
],
"b": [
0,
1,
0
]
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/algebra/cross-product",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": [
1,
0,
0
],
"b": [
0,
1,
0
]
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/cross-product", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":[1,0,0],"b":[0,1,0]}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":[1,0,0],"b":[0,1,0]}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/cross-product", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"a": [
1,
0,
0
],
"b": [
0,
1,
0
]
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "algebra.cross_product",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Limits
max_abs_component | 1000000000000000 |
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
What the cross product means and what the response contains
This calculator returns the normalized input vectors a and b, the three-component cross array, the non-negative area, a magnitude field equal to that area for callers that prefer a physics-style name, the method tag cross_product, the precision used, and a short formula reminder. There is no approximate Monte Carlo path and no symbolic CAS rewrite: the engine either builds exact floating-point components inside the supported range or rejects the input with invalid_input. Rejected cases include missing a or b, non-finite components, magnitudes above the documented ceiling, and any vector that is not three-dimensional, because the classical vector product is defined for length three and a length-two or length-four array would be a different operation.
How a and b are read and how results stay reproducible
Optional object forms with x y z keys are also accepted when a form posts structured components instead of a bare array. Optional leading plus signs and scientific notation on component strings are allowed when they parse as finite decimals; embedded spaces, hex prefixes, and non-numeric tokens are rejected with a clear message so a bad cell does not become a wrong normal. Zero vectors and parallel vectors are valid inputs: they simply yield a zero cross product and a zero parallelogram area, which is the correct geometric answer rather than an error. The preferred fields are a for the first vector and b for the second, each as an array of exactly three finite numbers, but aliases u and vector_a and left for a, and v and vector_b and right for b, are accepted so spreadsheets, query strings, and typed forms can reuse the same contract. Any vector whose length is not exactly three fails with invalid_input and a message that states the required three-dimensional shape, so a two-component planar pair or a four-component homogeneous coordinate tuple cannot silently pass. An optional precision field, with aliases decimals and dp, chooses how many decimal places to expose for each component and for the area, defaulting to ten and capped at fifteen so JSON goldens stay stable. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline without re-implementing the right-hand rule yourself.
Where cross products fit in study, physics, and software
On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split — free interactive exploration and a priced, identical API — matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result. Pair it with matrix multiplication or linear-system tools when you need related algebra steps, and with other math endpoints when you are building a small lab of free, deterministic vector helpers.
Integration use cases
Deterministic test fixtures
Generate reproducible cross and area fields for unit tests, mesh normal checks, and CI assertions.