# Dot product

For vectors a and b, multiply matching entries and add. It is zero exactly when the vectors are at right angles, and it equals |a||b| cos(angle).

Maths (university) · Part 08: Vectors, matrices and linear systems · Card 2 of 9 · Last updated 2026-10-19

## Analogy

Measuring how much two people are pushing a box in the same direction.

## Example

**(1, 2, 3) dot (4, -5, 6)**

1 x 4 = 4.
2 x (-5) = -10.
3 x 6 = 18.
Total 4 - 10 + 18 = 12.

## In practice

Used to find angles, projections and work done by a force.

## Practice

### Flashcards

- **What is (1, 2, 3) dot (4, -5, 6)?** 12.
- **What does a zero dot product tell you?** The two vectors are perpendicular.

## Questions people ask

### How do I use the dot product to find an angle?

Rearrange a dot b equals length of a times length of b times cos of the angle. So cos of the angle is the dot product divided by the two lengths. For (1, 0) and (1, 1), that is 1 over root 2, so 45 degrees.

### What does a dot product of zero mean?

The two vectors are perpendicular (orthogonal), assuming neither is the zero vector. For example, (2, 1) dot (1, -2) is 2 minus 2, which is zero. This is a quick test for right angles in geometry and physics. It saves you from drawing the vectors or measuring any angle with a protractor.

## Related concepts

- [Vector](https://studyvisually.com/learn/science/maths-university/maths-university-part-8/maths-university-part-8-concept-1)
- [Cross product](https://studyvisually.com/learn/science/maths-university/maths-university-part-8/maths-university-part-8-concept-3)

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