# ¶ Vectors

## ¶ Mathematical Vectors

A vector is a concept used to convey quantities that cannot be expressed by a single number.

Think about velocity, which is a mathematician's way of saying "speed plus direction."

Speed is 10 m/s. Velocity is 10 m/s in a north-west direction.

## ¶ Vector Notation

Vectors are incredibly versatile and show up again and again across mathematics. There are many different ways to express a vector, each having their own benefits and drawbacks.

Here are just a few of the ways we can express the same information.

## ¶ Multi-Dimensional Vectors

Below is an example of a 3-dimensional vector.

This is as far as the human brain can visualize, but mathematical dimensions can continue far beyond 3; they can go arbitrarily high.

A vector in n-dimensional space can hold up to n data points, one piece of data in each dimension.

The easiest way to see this is in the (v_0, v_1, ... , v_n) notation. Each dimension provides capacity to store an extra point of data.

Now you might be asking yourself "ok, I understand that we can put data into a vector, but why would I want to?"

Remember, vectors are mathematical primitives. They are incredibly powerful because they can be programmatically manipulated.

## ¶ Vectors in Computer Science

Think about 3D space, a vector just looks like an arrow.

Take two arrows, place them tip-to-tip. Draw a new arrow from the start of the first one to the end of the second.