Vectors: arrows with algebra

Everything in this course — matrices, gradients, the whole machinery of higher math — is built from one object. Not a number: an arrow.

Two faces of the same thing

A vector is simultaneously:

• Geometry: an arrow with a length and a direction. Where it starts doesn't matter — only where it points and how far.
• Algebra: a list of components, v = (3, 2), read as "3 across, 2 up."

The entire power of the subject is that you can switch faces at will: prove things with pictures, compute things with lists.

The two operations

Vectors come with exactly two moves, and both are picture-simple:

Addition — walk one arrow, then the other:

Component-wise it's even simpler: add the corresponding entries. The picture and the arithmetic always agree — that agreement is what makes the component representation legitimate.

Scaling — multiply by a number: 2v is twice as long, ½v half, and −v is the same arrow spun 180°. Numbers used this way are called scalars because all they do is scale.

Length

The length (or norm) of v = (x, y) is Pythagoras, nothing more:

$|v| = \sqrt{x^2 + y^2}$

A vector of length 1 is a unit vector — pure direction with the size stripped away. Divide any vector by its own length and you get its direction: v/|v|.

Why this matters

Addition and scaling look humble, but they define the whole game: anything you can build with those two moves (and that's a lot — forces, velocities, pixel colors, portfolio weights) obeys the same math. The next lesson adds the one product that measures how two vectors relate.