Vectors

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]
```

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.

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 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

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

Prerequisites