**Vectors, concretely, are arrows**, with a head and a tail. If two arrows share a tail, then you can measure the angle between them. The length of the arrow represents the magnitude of the vector.

The **modern abstract view** is much more interesting but let’s start at the beginning.

### Force vectors

Originally vectors were conceived as a **force applied at a point**.

As in, “That lawn ain’t mowing itself, boy. Now you git over there and apply a **continuous stream of vectors** to that lawnmower, before I apply a **high-magnitude vector** to your bee-hind!”

### Thanks Galileo, totally gonna get you back, man

The **Galilean idea** of splitting a point into its **x**-coordinate, **y**-coordinate and **z**-coordinate works with vectors as well. “**Apply a force** that totals 5 foot-pounds / second² in the **x** direction and 2 foot-pounds / second² in the **y** direction”, for instance.

Therefore, both points and vectors benefit from **adding more dimensions** to Galileo’s “coordinate system”. Add a **w** dimension, a **q** dimension, a **ξ** dimension — and **it’s up to you to determine** what those things **can mean**.

If a vector can be described as (5, 2, 0), then why not make a vector that’s (5, 2, 0, 1.1, 2.2, 19, 0, 0, 0, 3)? And so on.

### 4th Dimension Plus

So that’s how you get to 4-D vectors, 13-D vectors, and 11,929-D vectors. But the really interesting stuff comes from considering **∞-dimensional vectors**. That opens up** functional space**, and sooooo many things in life are functions.

(Interesting stuff also happens when you make vectors out of things that are not traditionally conceived to be “numbers”. Another post.)

## Abstractions

The modern, abstract view includes **as vectors**:

- lists
- 1-D arrays
**bitstreams**- linear functionals
- force vectors
- personal preferences
- decisions
- desires
**the flow of heat**- dinosaur tracks
- tying a knot
- economic transactions
- a story or article, in any language
- a poem
- water flowing
- time series
- one instant during an argument
- a curve
- probabilities associated with various outcomes
**marketing data**- statistical observations
- bids and asks
- the quantum numbers of an atom
- waveforms
- signals
- songs
**heartbeats**- one solution of a differential equation
**a polynomial**- a jpeg or bitmap of the Mona Lisa
- a set of instructions
- a dance move
- someone’s signature
**a secret message**- a Taylor series
- a Fourier decomposition
- turning your mattress

(which you’re apparently supposed to do once a season) - intentions
- electromagnetic flux
- part of a trajectory
- one wisp of the wind
**states of affairs**- logical propositions
- distortions in a crystal lattice
- a rotation of Rubik’s cube
- neuronal spike-trains — so, thoughts? perceptions?
**colour**

### Things you can do with vectors

Given two vectors, you should be able to take their outer product or their inner product.

The inner product allows you to measure the **angle between two vectors**. If the inner product makes sense, then the space you are playing in has geometry. (Not all spaces have geometry — some just have topology.)

And — this is weird — if the concept of angle applies, then the concept of **length** applies as well. Don’t ask me why; the symbols just work that way.

### Magnitude

But the “length” of a song (one of my for-instances above) would not be something like 2:43. **The magnitude of a song vector** would be the total amount of energy in the sound wave / compression wave.

What is the angle between two songs? **What is the angle between two heartbeats?**

I don’t know, but I could calculate it.

### Linear Algebra

Also, you can do **linear algebra** on vectors — provided they’re coming out of the same point. Some might say that the ability to do linear algebra on something is what makes a vector.

That can mean different things in different spaces — like maybe you’re **superposing wave-forms**, or maybe you’re **converting bitmap images to JPEG**. Or maybe you’re Photoshopping an existing JPEG. Oh, man, Photoshop is so math-y.

Shearing the mona lisa (linear algebra on an image — from the Wikipedia page on eigenvectors, which are the vectors shown)