# Vector Angle Calculator

> Find the angle between two vectors in degrees and radians from their dot product and magnitudes, and see whether they are perpendicular or parallel.

Category: Algebra, geometry & advanced math · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/vector-angle-calculator

The angle between two vectors comes from the dot product: cos θ = (A·B) ÷ (|A| |B|). Enter the components separated by commas, in any number of dimensions.

Also called: Angle between vectors calculator.

## Inputs
- Vector A (default 3, 4)
- Vector B (default 4, 3)

## Results
- Angle between the vectors
- Angle in radians
- cos θ
- Dot product A·B
- |A|
- |B|
- Relationship

## Formula

```
cos θ = (A·B) ÷ (|A| × |B|);  A·B = Σ aᵢbᵢ;  |A| = √(Σ aᵢ²);  θ = arccos(cos θ)
```

Where:
- A, B: the two vectors
- aᵢ, bᵢ: their components
- θ: the angle between them, from 0° to 180°

## Worked example
1. A·B = 3×4 + 4×3 = 24.
2. |A| = √(9+16) = 5 and |B| = 5.
3. cos θ = 24 ÷ 25 = 0.96, so θ = arccos 0.96 ≈ 16.2602°.

**Result:** The vectors (3, 4) and (4, 3) are about 16.26° apart.

## What the result means

A dot product of zero means a right angle; a positive one means less than 90° and a negative one more than 90°. The angle never exceeds 180° because it is measured the short way round.

## Assumptions and limits
- Both vectors are in the same space and use the same orthonormal (ordinary Cartesian) basis.
- The angle is undefined if either vector is zero.
- This is the unsigned angle; a signed (clockwise or anticlockwise) angle needs a 2D orientation convention.

## Questions

### Does the order of the vectors matter?

No. The dot product is symmetric, so the angle from A to B equals the angle from B to A.

### Why is cos θ clamped?

Rounding can push the ratio a hair above 1 for parallel vectors; it is limited to the range −1 to 1 so arccos stays defined.

## Related calculators
- [Vector Magnitude Calculator](https://calculators.geekmidst.com/calculators/vector-magnitude-calculator)
- [Unit Vector Calculator](https://calculators.geekmidst.com/calculators/unit-vector-calculator)
- [Triangle Calculator](https://calculators.geekmidst.com/calculators/triangle-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/vector-angle-calculator