Vector magnitude calculator
The vector (3, 4) has a magnitude of 5 and the unit vector (0.6, 0.8): enter the components in 1 to 256 dimensions and get the length, its square and the normalized direction. Free in your browser.
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The magnitude of a vector is the Euclidean length that turns a list of components into a single non-negative size: the square root of the sum of the squared components. In two dimensions that is the familiar hypotenuse of a right triangle; in three dimensions it is the space diagonal of a rectangular box; in higher dimensions the same formula continues without change. Dividing every component by that magnitude produces a unit vector that points in the same direction with length one, which is the standard way to normalize directions for graphics, physics, machine learning, and linear algebra homework. This calculator accepts a real vector of any dimension from one through two hundred fifty-six, returns the magnitude and the Euclidean norm as the same L2 value, the squared magnitude for callers that want to avoid a square root, and optionally the unit vector.
How Euclidean magnitude and unit vectors work
Start with a real vector v equals the ordered list of finite components v one through v n for some dimension n at least one. The Euclidean magnitude, also called the L2 norm, is defined by the formula magnitude equals the square root of the sum of each component squared. That quantity is always greater than or equal to zero, and it is zero if and only if every component is zero.
When the magnitude is strictly positive, the unit vector is obtained by dividing every component by the magnitude, which yields a vector of length one that preserves direction. Nested arrays are rejected so a matrix cannot be confused with a one-dimensional vector. Empty arrays and dimensions above two hundred fifty-six fail validation.
Where vector magnitude calculators show up in real work
Secondary-school and college linear algebra courses constantly assign magnitude drills, unit-vector normalizations, and direction cosines; a single trusted oracle that returns cleaned components and the L2 length speeds grading and demo setup. Physics and engineering pipelines normalize force, velocity, and displacement vectors before integrating motion or projecting onto axes, and they need an honest failure when a zero vector cannot define a direction.
Computer graphics and game engines convert arbitrary displacement vectors into unit directions for lighting, camera look-at, and collision response, often in two or three dimensions but sometimes in higher-dimensional configuration spaces. If you later need the distance between two planar points, matrix determinants, cross products in three dimensions, or Minkowski norms other than L2, treat those as separate search intents and landings rather than overloading this vector magnitude calculator page.
What you can do with it
Homework and exam answer keys
Generate magnitude, squared magnitude, and unit vectors for textbook n-tuples so teachers and auto-graders share one ground truth.
Physics and engineering direction vectors
Normalize force, velocity, or displacement vectors to unit length and reject the zero vector when a direction is required.
Feature and embedding normalization
Scale real feature rows to Euclidean length one before similarity, clustering, or distance metrics in data pipelines.
FAQ
What is the difference between magnitude and Euclidean norm?
For real vectors they are the same L2 quantity: the square root of the sum of squared components. This calculator returns both fields with identical cleaned values for naming convenience.
Which dimensions are supported?
Any dimension from 1 through 256 inclusive. Empty arrays and nested (matrix-shaped) arrays are rejected.
How is the unit vector computed?
The result has Euclidean length one and the same direction as the input when the input is non-zero.
What does it cost?
The browser tool is free and runs locally.
How do I find the magnitude of the vector (3, 4)?
Square each component, add the squares and take the square root. For (3, 4) the squares add up to 25, so the magnitude is 5, and dividing each component by 5 gives the unit vector (0.6, 0.8).
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/vector-magnitude \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"vector":[3,4]}'const res = await fetch("https://api.kit.forhosting.com/algebra/vector-magnitude", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"vector": [
3,
4
]
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/algebra/vector-magnitude",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"vector": [
3,
4
]
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/vector-magnitude", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"vector":[3,4]}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"vector":[3,4]}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/vector-magnitude", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"vector": [
3,
4
]
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "algebra.vector_magnitude",
"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.
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
When the input is the zero vector and a unit vector is requested, the request is rejected with a clear validation error instead of dividing by zero.
How Euclidean magnitude and unit vectors work
The same number is often written with double vertical bars as the Euclidean norm of v, so this capability reports magnitude and euclidean_norm as identical cleaned values. When the magnitude is zero, no unit vector exists on the real line, because division by zero is undefined; the calculator therefore rejects the zero vector whenever a unit vector is requested, and it explains that callers may set want_unit to false if they only need the zero magnitude. The pipeline is deterministic: no random seeds, no wall-clock timestamps, no network calls, and no model drift, so identical components always produce identical JSON. Missing fields, non-numeric strings, NaN, and Infinity fail with invalid_input and are not charged.
What you send and what you get back
Textbook checks include the two-dimensional case three four with magnitude five and unit vector zero point six and zero point eight; the three-dimensional case one two two with magnitude three; and the one-dimensional case negative seven with magnitude seven and unit vector negative one. Coefficients must be finite; dimensions outside one through two hundred fifty-six are rejected before any arithmetic runs. Send the field vector as an array of finite numbers, for example three and four for the classic right triangle with magnitude five, or a JSON string of the same array shape when the client only speaks text. The optional boolean want_unit defaults to true and controls whether the unit vector is computed. When want_unit is true and every component is zero, the request fails with invalid_input and a message that the vector is all zeros. When want_unit is false, the zero vector is allowed and returns magnitude zero with unit_vector set to null. On success the response echoes the cleaned vector, the integer dimension, magnitude, euclidean_norm, squared_magnitude, is_zero, want_unit, unit_vector when requested, and a short formula string. Pricing for successful API calls is the published base of $0.002 per item; validation failures do not consume prepaid balance.
Where vector magnitude calculators show up in real work
Curriculum authors building interactive lessons want the same JSON shape in the free tool and the API so offline worksheets and online auto-graders stay aligned. Because this capability is a pure parse engine, API pricing is the published base of $0.002 per successful item and failed validation does not consume balance.
Technical questions
What happens if the vector is all zeros?
If want_unit is true (the default), the request fails with invalid_input because a unit vector cannot be defined. Set want_unit to false to receive magnitude 0 without a unit vector.