#### Definition

- Vectors are mathematical objects which represent change.
- They are the complement to coordinates, which represent fixed points.
- They have two properties:
- Direction
- Magnitude (or length)

- They are often written as how much change they apply in each dimension:
- A two dimensional vector that changes 3 in the first dimension and 5 in the second dimension is written as
**[3, 5]**

- When writing a variable which holds a vector, an arrow is placed above its symbol:
LaTeX: `\vec a = [3, 5]`

#### Vectors in one dimension

- A vector with only one dimension is written as a single value enclosed in a set of square brackets:
- This can also be represented graphically:
- Magnitude:
- The magnitude of a one dimensional vector is simply the absolute value of its only element.
- The vectors
`[5]` and `[-5]` both have a magnitude of 5

- Direction:
- A one dimensional vector can only have two possible directions: forwards and backwards.
- The vectors
`[5]` and `[-5]` have the same magnitude but different directions. - The vectors
`[-3]` and `[-5]` have different magnitudes but the same direction.

#### Vectors in two dimensions

- A vector with two dimensions is written as a pair of values enclosed in a set of square brackets:
- This can also be represented graphically:
- Magnitude:
- The magnitude of a two dimensional vector can be calculated using the Pythagorean theorem.
- The two components of the vector form two sides of a right triangle.
- The magnitude of the vector is equal to the hypotinuse.
- For example:
- If we apply the Pythagorean theorem:

#### Vectors in more than two dimensions

- Vectors can be defined in as many dimensions as desired.
- For example, a 5 dimensional vector would look like this:
- Magnitude:
- To calculate the magnitude of a vector with
`n` dimensions we use the following formula: - For example, the magnitude of: is

#### code (Python)

In python, vectors are represented as arrays.```
import math
import numpy as np
vector = [3, 6, 2]
magnitude = math.sqrt(np.dot(vector, vector))
print(magnitude) # prints: 7.0
```