# Dot Product and Angle Between Vectors Calculator

URL: https://speedycalculator.com/calculators/math/dot-product/
Last updated: 2026-10-25

> This calculator finds the dot product of two 3D vectors and the angle between them. It also tells you whether the vectors are perpendicular.

## Inputs

- a₁
- a₂
- a₃
- b₁
- b₂
- b₃

## Results

- Dot product
- |a|
- |b|
- Angle between vectors
- Cross product x
- Cross product y
- Cross product z

## Formula

```
a · b = a₁b₁ + a₂b₂ + a₃b₃
θ = arccos(a · b ÷ (|a||b|))
```

## Explanation

The dot product measures how much two vectors point in the same direction. Zero means they are perpendicular.

### How to use the Dot Product and Angle Between Vectors Calculator

- **a₁**
- **a₂**
- **a₃**
- **b₁**
- **b₂**
- **b₃**

Results update as you type, and every field accepts the units shown in its menu.

### How it works

Multiply matching components and add them. Divide by both magnitudes and take the arccosine for the angle.

### Inputs explained

FieldWhat to entera₁A numbera₂A numbera₃A numberb₁A numberb₂A numberb₃A number

### What each result means

- **Dot product** = [a₁]×[b₁] + [a₂]×[b₂] + [a₃]×[b₃]
- **|a|** = sqrt([a₁]^2 + [a₂]^2 + [a₃]^2)
- **|b|** = sqrt([b₁]^2 + [b₂]^2 + [b₃]^2)
- **Angle between vectors** = deg(acos([Dot product]÷([|a|]×[|b|])))
- **Cross product x** = [a₂]×[b₃] - [a₃]×[b₂]
- **Cross product y** = [a₃]×[b₁] - [a₁]×[b₃]
- **Cross product z** = [a₁]×[b₂] - [a₂]×[b₁]

### Example values

How the main result changes when **a₁** changes and everything else stays at the example values above:

a₁Dot product0.5100.75111121.25131.514216

## Worked example

Inputs — a₁: 1; a₂: 2; a₃: 3; b₁: 4; b₂: -5; b₃: 6.

Results — **Dot product = 12**; |a| = 3.741657; |b| = 8.774964; Angle between vectors = 68.5624 °; Cross product x = 27; Cross product y = 6; Cross product z = -13.

## FAQ

**What does a negative dot product mean?**
The angle between the vectors is more than 90°.

**What is the cross product?**
A vector perpendicular to both, also shown here.

**What formula does the Dot Product and Angle Between Vectors Calculator use?**
a · b = a₁b₁ + a₂b₂ + a₃b₃.