# Vector

A quantity with a size and a direction, written as a list of numbers such as (3, 4). Its length is the square root of the sum of squares of its entries.

Maths (university) · Part 08: Vectors, matrices and linear systems · Card 1 of 9 · Last updated 2026-10-19

## Analogy

A set of walking directions: 3 steps east and 4 steps north.

## Example

**Vector v = (3, 4)**

Length = sqrt(3^2 + 4^2).
= sqrt(9 + 16) = sqrt(25).
Length = 5.
2v = (6, 8), length 10.

## In practice

Vectors describe forces, velocities and data points with many features.

## Practice

### Flashcards

- **What is the length of the vector (3, 4)?** 5.

## Questions people ask

### How do I add and scale vectors?

Add vectors entry by entry, so (1, 2) plus (3, 4) is (4, 6). To scale, multiply every entry by the number, so 2 times (3, 4) is (6, 8). Scaling by a positive number keeps the direction and changes the length. Scaling by a negative number also flips the arrow around to point the opposite way.

### What is a unit vector?

A unit vector has length 1 and shows pure direction. To make one, divide a vector by its length. For (3, 4), the length is 5, so the unit vector is (3/5, 4/5), which is (0.6, 0.8). Its length is sqrt(0.36 + 0.64) = 1.

## Related concepts

- [Dot product](https://studyvisually.com/learn/science/maths-university/maths-university-part-8/maths-university-part-8-concept-2)
- [Cross product](https://studyvisually.com/learn/science/maths-university/maths-university-part-8/maths-university-part-8-concept-3)

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