Euclidean norm
High school
Consider a vector $x = (x_1, \dots, x_n)$, drawn as an arrow from the origin. The Euclidean norm of $x$ is the length of the arrow:
\[\norm{x}_2 = \sqrt{x_1^2 + \dots + x_n^2}.\]Examples
Worked examples
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The vector $x = (1, 1) \in \reals^2$
: $\norm{x}_2 = \sqrt{1^2 + 1^2} = \sqrt{2}$
High school
The three norms of $(1, 1)$ are as far apart as possible for a vector in $\reals^2$: the 1-norm is $\sqrt{2}$ times the Euclidean norm, which is $\sqrt{2}$ times the infinity norm.
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The vector $x = (3, -4) \in \reals^2$
: $\norm{x}_2 = \sqrt{3^2 + (-4)^2} = 5$
High school
The Euclidean norm of $(3, -4)$ is the length of the arrow from the origin, $5$.
-
The vector $x = (5, 0) \in \reals^2$
: $\norm{x}_2 = \sqrt{5^2 + 0^2} = 5$
High school
All three norms of $(5, 0)$ equal $5$, because the vector lies along an axis: the norms differ only for vectors that point between the axes.
Properties
- The Euclidean norm comes from the dot product, as $\norm{x}_2 = \sqrt{x \cdot x}$, so it is the only one of the common norms on $\reals^n$ that measures angles as well as lengths.
- Rotations and reflections do not change the Euclidean norm of a vector.
Notes
When no norm is specified, $\norm{x}$ on $\reals^n$ usually means the Euclidean norm.
Distinctions involving this term
- Types of norms: A norm measures the size of a vector, and $\reals^n$ has several common norms that measure size in different ways. The Euclidean norm is the straight-line length, the 1-norm adds the sizes of the entries, and the infinity norm takes the largest entry. Each is a norm, so each satisfies the triangle inequality, and for every vector, $\norm{x}_{\infty} \leq \norm{x}_2 \leq \norm{x}_1$.