What is a Vector?

A vector is an arrow with direction and magnitude.

Components

In 2D, a vector v has two components:

• v₁ — horizontal (x-axis)
• v₂ — vertical (y-axis)

We write: v = (v₁, v₂)

$$\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$$

The first component $v_1$ tells you how far to go horizontally, and $v_2$ tells you how far to go vertically.

Magnitude

The magnitude (or length) of a vector is:

$$\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2}$$

This is just the Pythagorean theorem!

Direction

The direction of a vector is the angle it makes with the positive x-axis:

$$\theta = \arctan\left(\frac{v_2}{v_1}\right)$$