# Vector Calculator

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

> The vector calculator performs the common operations on 2D or 3D vectors: addition, subtraction, dot product, angle between vectors, normalisation, projection and the vector between two points.

## Inputs

- Vectors in
- Operation
- a – x
- a – y
- a – z
- b – x
- b – y
- b – z

## Results

- Result vector – x
- Result vector – y
- Result vector – z (3D)
- Dot product a · b
- Angle between a and b
- Length of the result

## Formula

```
a ± b = (aₓ ± bₓ, a_y ± b_y, a_z ± b_z)
a · b = Σ aᵢbᵢ,  proj_a b = (a·b / |a|²) a
```

## Explanation

Vectors describe quantities with size and direction – displacement, velocity, force. These operations are the toolkit of mechanics, computer graphics and navigation.

### How to use the Vector Calculator

- **Vectors in**
- **Operation**
- **a – x**
- **a – y**
- **a – z**
- **b – x**
- **b – y**
- **b – z**

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

### How it works

Choose the operation; the relevant results appear. For "vector from two points", a is the starting point and b the end point.

### Tips

- Sketch the points first – a quick drawing shows whether a slope should be positive or negative.
- Keep the order of the points consistent in every formula.
- Vertical lines have an undefined slope; horizontal lines have slope zero.
- Use the same units on both axes when you measure distances or angles.

### Common mistakes to avoid

- Swapping x and y coordinates.
- Subtracting the coordinates in a different order in the numerator and denominator of the slope.
- Forgetting that the distance is always positive.

### Inputs explained

FieldWhat to enterVectors inChoose one: 2D, 3DOperationChoose one: Addition a + b, Subtraction a − b, Dot product, Angle between vectors …a – xA numbera – yA numbera – zA numberb – xA numberb – yA numberb – zA number

### Example values

How the main result changes when **a – x** changes and everything else stays at the example values above:

a – xResult vector – x0.54.50.754.75151.255.251.55.526

## Worked example

Inputs — Vectors in: 3D; Operation: Addition a + b; a – x: 1; a – y: 2; a – z: 3; b – x: 4; b – y: 0; b – z: -2.

Results — **Result vector – x = 5**; Result vector – y = 2; Result vector – z (3D) = 1; Length of the result = 5.47722558.

## Key terms

- **Vector**: A quantity with magnitude and direction.

## FAQ

**How do I add vectors?**
Add their corresponding components.

**What does the dot product tell me?**
How aligned two vectors are; zero means perpendicular.

**What is normalisation?**
Dividing a vector by its length to get a unit vector.

**How do I find the vector from P to Q?**
Subtract: Q − P.

**What formula does the Vector Calculator use?**
a ± b = (aₓ ± bₓ, a_y ± b_y, a_z ± b_z).


## Sources

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