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Vector Subtraction Calculator | Difference from components

Subtract 2D or 3D vectors component by component, then get the resulting magnitude and direction angle with the working shown. Essential for physics displacement problems, engineering and graphics work.

Last updated: June 2026 · Free · No sign-up required

Results

Enter values above and click Calculate to see your result instantly.

Inside the Vector Subtraction Calculator

The engine behind this calculator evaluates A − B = (aₓ − bₓ, aᵧ − bᵧ); |R| = √(x²+y²); θ = atan2(y, x) directly in your browser — nothing is sent to a server, and results update as you type.

For context, the sections beneath the calculator include typical values, a worked example you can recompute by hand, and an FAQ covering the practical details of vector subtraction calculator.

Quick reference

ABA − B|A − B|
(5, 3)(2, 1)(3, 2)3.606
(1, 0)(0, 1)(1, −1)1.414
(4, 4)(4, 4)(0, 0)0

info Vector Subtraction Calculator

Free math calculator — enter your numbers and get an instant, accurate result.

info Private by design

Everything runs locally in your browser. No uploads, no accounts, no tracking.

info Works everywhere

Fully responsive and mobile-friendly — calculate on any device, any time.

info Educational

Includes the formula and step-by-step explanation so you understand the math, not just the answer.

Interpretation guide

ResultMeaning
A − BThe vector from point B to point A
MagnitudeDistance between the two tips
DirectionAngle of that displacement from +x

lightbulb Worked example

Let's say you are using the Vector Subtraction Calculator. Calculates vector subtraction using A − B = (aₓ − bₓ, aᵧ − bᵧ), live in your browser. Enter the values that match your situation into the input fields and press calculate — using realistic numbers makes the result directly useful for you.

Result: The calculator instantly applies the formula A − B = (aₓ − bₓ, aᵧ − bᵧ); |R| = √(x²+y²); θ = atan2(y, x) and returns the result with appropriate precision.

What this means: Read the result in the context of what you are measuring. The step-by-step breakdown lets you confirm the math and understand which input most affects the outcome.

Formula & methodology

Formula: A − B = (aₓ − bₓ, aᵧ − bᵧ); |R| = √(x²+y²); θ = atan2(y, x)

The Vector Subtraction Calculator is built on a well-established calculation method. It uses the formula A − B = (aₓ − bₓ, aᵧ − bᵧ); |R| = √(x²+y²); θ = atan2(y, x) to turn your inputs into a reliable result. Calculates vector subtraction using A − B = (aₓ − bₓ, aᵧ − bᵧ), live in your browser. The steps are shown on the page so you can follow the reasoning from input to output.

  1. Enter the x and y components of vectors A and B.
  2. Subtract component-wise: R = (aₓ − bₓ, aᵧ − bᵧ).
  3. Magnitude |R| = √(x² + y²); direction θ = atan2(y, x), measured from the +x axis.

Example: (5, 3) − (2, 1) = (3, 2); |R| = √13 ≈ 3.6056 at 33.69°.

Authoritative source: Wolfram MathWorld

Frequently asked questions

How do you subtract vectors?
Enter the required values into the input fields and press the calculate button. The tool applies the formula A − B = (aₓ − bₓ, aᵧ − bᵧ); |R| = √(x²+y²); θ = atan2(y, x) and returns the result with a short explanation of each step, so you can check the working yourself.
What does A − B represent?
The Vector Subtraction Calculator is a free online tool that calculates vector subtraction using a − b = (aₓ − bₓ, aᵧ − bᵧ), live in your browser. You enter a few values, and it returns an accurate result together with a step-by-step explanation, so it works as both a calculator and a quick reference for the underlying concept.
Is vector subtraction commutative?
The Vector Subtraction Calculator is free, private, and accurate: it runs entirely in your browser (no uploads, no accounts), applies the standard calculation, and explains each step so you can verify the result. Calculates vector subtraction using A − B = (aₓ − bₓ, aᵧ − bᵧ), live in your browser. There is no limit on usage, and it works on any device.
How do I get magnitude and direction?
After subtracting component-wise, the magnitude is √(x² + y²) for 2D (with the z term in 3D), and the direction is the angle θ = atan2(y, x) measured from the positive x-axis. The calculator returns both automatically, so a result vector of (3, −4) gives magnitude 5 and direction about −53.1°.
Does this handle 3-D vectors?
Yes — the calculator accepts three-component vectors and subtracts them component-wise: (x₁−x₂, y₁−y₂, z₁−z₂). The magnitude extends to √(x² + y² + z²), and the direction is expressed via the angles with each axis. 3D vector subtraction is common in physics, graphics, and engineering displacement problems.
What does the Vector Subtraction Calculator do?
Calculates vector subtraction using A − B = (aₓ − bₓ, aᵧ − bᵧ), live in your browser. The calculator takes your inputs, applies the standard calculation, and returns a clear result so you can make an informed decision without doing the math by hand.
What formula does the Vector Subtraction Calculator use?
The Vector Subtraction Calculator applies the standard formula: A − B = (aₓ − bₓ, aᵧ − bᵧ); |R| = √(x²+y²); θ = atan2(y, x). The tool walks through each step of the calculation so you can verify the numbers yourself.
Is the Vector Subtraction Calculator free and private?
Yes — the Vector Subtraction Calculator. There is no sign-up, no paywall, no trial, and no limit on how many calculations you can run. CalcaTools covers its costs with non-intrusive display advertising, so the calculator itself never asks for payment or restricts any feature.
How accurate is the Vector Subtraction Calculator?
The Vector Subtraction Calculator applies the standard formula: A − B = (aₓ − bₓ, aᵧ − bᵧ); |R| = √(x²+y²); θ = atan2(y, x). It is tested against published worked examples before launch and uses native double-precision arithmetic, so results are accurate to typical precision for the values you enter — with no intermediate rounding that could distort the answer.

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