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Vector Subtraction Calculator | Difference from components

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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

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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.

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

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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: plane velocity minus wind

A pilot's airspeed vector is A = (150, 40) knots (east, north components) and the wind vector is B = (20, −10) knots. Ground velocity = A − B. Subtracting component-wise: Rx = 150 − 20 = 130, Ry = 40 − (−10) = 50. Magnitude |R| = √(130² + 50²) = √(16900 + 2500) = √19400 ≈ 139.28. Direction θ = atan2(50, 130) ≈ 21.04°.

Result: Ground velocity = (130, 50) knots — speed ≈ 139.3 knots on a heading ≈ 21.0° north of east.

What this means: The wind effectively pushes the plane 20 knots westward and 10 knots southward relative to its airspeed, so the actual track over the ground is slower and angled north of the intended heading. Subtracting the wind vector isolates what the ground — and air-traffic control — actually sees.

Formula & methodology

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

How vector subtraction is computed

Vectors subtract component by component. For A = (aₙ, aᵧ) and B = (bₙ, bᵧ):

R = A − B = (aₙ − bₙ, aᵧ − bᵧ)

  • Subtract the x-components and the y-components separately (and z-components for 3D vectors).
  • Magnitude: |R| = √(x² + y²) — the straight-line length of the result.
  • Direction: θ = atan2(y, x), in degrees measured from the +x axis.

What the result means

A − B points from the tip of B to the tip of A: it is the displacement you must add to B to reach A. Reversing the order flips the direction (B − A = −(A − B)), which is why subtraction, unlike addition, depends on order.

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.
How do you subtract vectors in a navigation problem?
Subtract component by component: relative velocity = your velocity − the wind's velocity. If a plane flies at (150, 40) knots east/north and the wind blows (20, −10), the ground velocity is (150−20, 40−(−10)) = (130, 50) knots. The magnitude and heading of that result vector then give the true speed and track.
What is the geometric meaning of A − B?
A − B is the vector that starts at the tip of B and ends at the tip of A — the displacement you must add to B to reach A. Its length √(x²+y²) is the straight-line distance between the two endpoints, and atan2(y, x) gives its direction measured from the +x axis.
Does the calculator handle 3D vectors?
Yes — for 3D vectors it subtracts the z-components too: (aₓ−bₓ, aᵧ−bᵧ, az−bz). The magnitude becomes √(x²+y²+z²), and the direction is reported against the coordinate planes. Each component subtraction is listed separately so the arithmetic stays checkable.
How is vector subtraction different from vector addition?
Subtraction is addition of the negated vector: A − B = A + (−B), where −B flips B's direction but keeps its length. That is why A − B ≠ B − A — reversing the order flips the result's sign, giving the opposite direction with the same magnitude.

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