Direction angle of a vector calculator
Get the direction of a 2D vector from its components: the vector (3, 4) points at 53.130102354156° from the positive x-axis, or 0.927295218002 radians, with a length of 5. The correct quadrant is found every time. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- X
- 3
- Y
- 4
- Magnitude
- 5
- Angle rad
- 0.927295218
- Angle deg
- 53.1301023542
- Angle deg 0 360
- 53.1301023542
The direction angle of a vector calculator finds the polar angle of a two-dimensional vector from its Cartesian components. You enter the horizontal component x and the vertical component y; the engine evaluates the two-argument arctangent atan2 of y and x so the result lands in the correct quadrant, reports the angle in radians and degrees, normalizes a copy into the half-open interval from zero to three hundred sixty degrees, and returns the Euclidean magnitude of the vector. No random sampling, no network call, and no unit conversion of the components themselves—callers choose the shared Cartesian frame.
What the direction angle means and what you get
In plane analytic geometry every non-zero vector has a unique direction measured from a fixed reference ray. Standard mathematical convention takes the positive x-axis as that reference and measures the direction angle counterclockwise for positive angles and clockwise for negative angles when the signed atan2 form is used. Equivalently, the same orientation can be written as a single turn in degrees from zero up to but not including three hundred sixty.
The classic one-argument arctangent of y over x is insufficient: it collapses opposite quadrants and fails when x is zero. The two-argument function atan2 of y and x fixes both problems by inspecting the signs of both components, so a vector in the second quadrant is not mistaken for one in the fourth, and pure vertical vectors on the positive or negative y-axis receive the correct ninety-degree or negative ninety-degree answers.
This calculator echoes the cleaned components, the magnitude equal to the square root of x squared plus y squared, the signed angle in radians and degrees, the normalized degree form in zero to three hundred sixty, a direction label naming the mathematical quadrant or the axis when a component is exactly zero, and a short formula string restating the identities.
Related tools such as vector magnitude, dot product, or slope from two points answer different search intents; the intent here is specifically the polar direction of one 2D vector from its components.
Where 2D direction angles fit in study, design, and automation
Students meet the direction angle when polar form is introduced: a textbook may give components of a force, a displacement, or a velocity and ask for the angle from the positive x-axis in degrees or radians. This calculator returns both unit systems plus the magnitude so a single call can check a full polar-form answer key.
Game developers and robotics hobbyists often need the heading of a velocity or aim vector on a 2D plane; converting components through atan2 without hand-rolling quadrant logic avoids silent sign bugs when the character faces left or down. Survey sketch tools, CAD macros, and physics notebooks reuse the same identity when a direction of travel or thrust must be recovered from measured Cartesian parts.
Pair it with vector magnitude when you need the length alone, with angle converters when results must appear in gradians or turns, and with graphing notes that show the arrow from the origin so students see that the number and the figure describe the same direction.
What you can do with it
Game headings and aim vectors
Recover the on-plane heading of a velocity or aim vector from vx and vy with correct quadrants for left, down, and diagonal motion.
FAQ
What does the direction angle of a vector calculator cost?
It is free in your browser on this page.
How is the direction angle calculated?
θ = atan2(y, x) from the positive x-axis. The response includes radians, signed degrees, degrees normalized to [0, 360), the magnitude √(x²+y²), and a quadrant or axis label.
What happens if the vector is zero?
When both x and y are zero the vector has no direction.
Why use atan2 instead of atan(y/x)?
atan(y/x) loses quadrant information and fails when x is zero. atan2(y, x) uses both signs so second and third quadrant vectors stay correct.
How is a negative angle turned into a value from 0 to 360 degrees?
Besides the signed angle, you get a second value in degrees from 0 up to just below 360, found by adding a full turn to negative results. A vector pointing down and to the right shows both forms.
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/geom/vector-angle-2d \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"x":3,"y":4}'const res = await fetch("https://api.kit.forhosting.com/geom/vector-angle-2d", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"x": 3,
"y": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/geom/vector-angle-2d",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"x": 3,
"y": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/vector-angle-2d", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"x":3,"y":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"x":3,"y":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/vector-angle-2d", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"x": 3,
"y": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "geom.vector_angle_2d",
"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
Use it free on this page or at $0.002 per successful API request when you need reproducible direction angles inside scripts and tests.
What the direction angle means and what the response contains
The zero vector is rejected with invalid_input because a displacement of length zero does not point anywhere; that discipline keeps homework checks honest and prevents pipelines from caching a silent zero-degree answer for a null input.
How the components are read and how the angle is computed
The preferred fields are x and y as JSON numbers; decimal strings are accepted for form posts and spreadsheet exports. Aliases vx and x_component map to x; aliases vy and y_component map to y. Both fields are required and must be finite—missing values, empty strings, NaN, and Infinity all raise invalid_input with a message that names the failing field. Computation is elementary and fully deterministic: the magnitude is the Euclidean hypotenuse of x and y; if that length is effectively zero the request fails; otherwise the angle in radians is Math.atan2 of y and x, the degree form multiplies by one hundred eighty over pi, and a second degree field normalizes the signed result into the half-open interval from zero to three hundred sixty by reducing modulo three hundred sixty and lifting negative remainders. The direction label reports positive_x_axis, negative_x_axis, positive_y_axis, or negative_y_axis when one component is exactly zero, and otherwise quadrant_1 through quadrant_4 in the standard mathematical sense. There is no third component, no 3D spherical angles, and no automatic conversion from degrees into gradians or turns; pair this endpoint with a dedicated angle converter when you need other units.
Where 2D direction angles fit in study, design, and automation
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 free-plus-priced split 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.
Integration use cases
Homework and polar-form checks
Convert textbook vector components into direction angle and magnitude, and confirm that the zero vector fails with invalid_input instead of a fake 0° answer.
Deterministic geometry fixtures
Generate reproducible atan2 angles in radians and degrees for unit tests, contest validators, and CI assertions without hand-rolled edge cases.
Technical questions
Which angle range is returned?
angle_rad and angle_deg follow the signed atan2 range (−π, π] and about (−180°, 180°]. angle_deg_0_360 is the same orientation normalized into [0, 360).