# Vector Addition Calculator

URL: https://speedycalculator.com/calculators/math/vector-addition/
Last updated: 2026-10-30

> The vector addition calculator adds or subtracts two vectors – optionally multiplied by scalars α and β – in 2D or 3D, entered as components or (in 2D) as magnitude and angle.

## Inputs

- Vectors in
- Operation
- Vector representation (2D)
- Multiple α (for a)
- Multiple β (for b)
- a – x (or magnitude)
- a – y (or angle °)
- a – z
- b – x (or magnitude)
- b – y (or angle °)
- b – z

## Results

- Result – x
- Result – y
- Result – z
- Magnitude of the result
- Direction of the result (2D)

## Formula

```
α·a ± β·b = (αaₓ ± βbₓ, αa_y ± βb_y, αa_z ± βb_z)
Polar input: x = m cos θ, y = m sin θ
```

## Explanation

Adding vectors combines displacements, forces and velocities: a plane's airspeed plus the wind gives its ground speed. The parallelogram (or tip-to-tail) rule is the geometric picture of component-wise addition.

### How to use the Vector Addition Calculator

- **Vectors in**
- **Operation**
- **Vector representation (2D)**
- **Multiple α (for a)**
- **Multiple β (for b)**
- **a – x (or magnitude)**
- **a – y (or angle °)**
- **a – z**
- **b – x (or magnitude)**
- **b – y (or angle °)**
- **b – z**

Results update as you type, and everything is calculated in your browser.

### How it works

Vectors given as magnitude and angle are converted to components first. The result is shown as components, magnitude and direction.

### Inputs explained

FieldWhat to enterVectors inChoose one: 2D, 3DOperationChoose one: Addition, SubtractionVector representation (2D)Choose one: Cartesian coordinates, Magnitude and angleMultiple α (for a)A numberMultiple β (for b)A numbera – x (or magnitude)A numbera – y (or angle °)A numbera – zA numberb – x (or magnitude)A numberb – y (or angle °)A numberb – zA number

### Example values

How the main result changes when **Multiple α** changes and everything else stays at the example values above:

Multiple α (for a)Result – x0.52.50.753.25141.254.751.55.527

## Worked example

Inputs — Vectors in: 2D; Operation: Addition; Vector representation (2D): Cartesian coordinates; Multiple α (for a): 1; Multiple β (for b): 1; a – x (or magnitude): 3; a – y (or angle °): 4; b – x (or magnitude): 1; b – y (or angle °): -2.

Results — **Result – x = 4**; Result – y = 2; Magnitude of the result = 4.47213595; Direction of the result (2D) = 26.5651 °.

## Key terms

- **Resultant**: The vector sum of several vectors.

## FAQ

**How do I add vectors graphically?**
Place them tip to tail; the sum goes from the first tail to the last tip.

**Is vector addition commutative?**
Yes, a + b = b + a.

**How do I subtract vectors?**
Add the opposite: a − b = a + (−b).

**What do α and β do?**
They scale each vector before adding – a linear combination.


## Sources

- [OpenStax – Calculus Volume 3](https://openstax.org/details/books/calculus-volume-3)
- [Wolfram MathWorld – Vector](https://mathworld.wolfram.com/Vector.html)