Plane Norm

Why

We want to talk about the size of a displacement.¹

Definition

The norm of a vector $x \in \R ^2$ is

\[ \sqrt{x_1^2 + x_2^2}. \]

Notation

We denote the norm of $x$ by $\norm{x}$. In other words, $\norm{\cdot}: \R ^2 \to \R $ is a function from vectors in $\R ^2$ to real numbers. The notation follows the notation of absolute value, the magnitude of a real number, and the double verticals remind us that $x$ is a vector. A warning: some authors write $\abs{x}$ for the norm of $x$ when it is understood that $x \in \R ^2$.

Visualization

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