Vectors in the Plane and Space

Vectors and Basic Operations

A vector is a quantity with magnitude and direction, represented by an arrow. In coordinates, a 2D vector is $(v_1, v_2)$ and a 3D vector is $(v_1, v_2, v_3)$; generally $\mathbb{R}^n$ uses $n$ components. Components are signed displacements along axes.

Two operations govern vectors:

Magnitude and Normalization

The magnitude (length or norm) is calculated via the Pythagorean theorem: $\|v\| = \sqrt{v_1^2 + v_2^2}$ in the plane, extending to space as $\|v\| = \sqrt{v_1^2 + v_2^2 + v_3^2}$. A unit vector has length 1. Normalizing a nonzero vector divides it by its length: $\frac{v}{\|v\|}$.

Common pitfall: Never add vectors by adding their magnitudes, and never confuse magnitude with components. Vectors add componentwise. The triangle inequality states $\|u + v\| \le \|u\| + \|v\|$, with equality holding only when $u$ and $v$ point in the exact same direction.