Normalize

v divided by its Euclidean norm -- a unit vector in v's direction.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageNormalize!↗

Normalize(v)v / Sqrt(Total(Abs(v)^2))

Normalize(v, f)v / f(v), a custom norm function

Typed signatures

Type	Package	Applies
(any, ((any) -> any)?) -> any	combinatorics	always

Domain: Collections

Details

• The zero vector is returned unchanged -- there is no direction to normalize it to.

Examples

Basic #

The 3-4-5 triangle's direction vector, normalized to unit length

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In$\mathrm{Normalize}(\bigl\lbrack3, 4\bigr\rbrack)$Out$\bigl\lbrack\frac{3}{5}, \frac{4}{5}\bigr\rbrack$

Divides by the Euclidean norm $\sqrt{27} = 3\sqrt3$, keeping exact radicals

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In$\mathrm{Normalize}(\bigl\lbrack1, 5, 1\bigr\rbrack)$Out$\bigl\lbrack\frac{\sqrt{3}}{9}, \frac{5\sqrt{3}}{9}, \frac{\sqrt{3}}{9}\bigr\rbrack$

Scope #

The zero vector has no direction, so it is left unchanged

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In$\mathrm{Normalize}(\bigl\lbrack0, 0\bigr\rbrack)$Out$\bigl\lbrack0, 0\bigr\rbrack$

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In$\mathrm{Normalize}(\bigl\lbrack x, y\bigr\rbrack, f)$Out$\bigl\lbrack\frac{x}{f(\bigl\lbrack x, y\bigr\rbrack)}, \frac{y}{f(\bigl\lbrack x, y\bigr\rbrack)}\bigr\rbrack$

live ↗

In$\mathrm{Normalize}(\bigl\lbrack x, y\bigr\rbrack)$Out$\bigl\lbrack\frac{x}{\sqrt{\vert x\vert^2+\vert y\vert^2}}, \frac{y}{\sqrt{\vert x\vert^2+\vert y\vert^2}}\bigr\rbrack$

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In$\mathrm{Normalize}(1+x+x^2, \operatorname{\_1}↦\int_{-1}^{1}\!\operatorname{\_1}^2\, \mathrm{d}x)$Out$\frac{1}{22}×5(x^2+x+1)$

live ↗

In$\mathrm{Normalize}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack4, 5\bigr\rbrack\bigr\rbrack, \mathrm{Norm})$Out$\bigl\lbrack\bigl\lbrack\frac{\sqrt{46}}{46}, \frac{\sqrt{46}}{23}\bigr\rbrack, \bigl\lbrack\frac{2\sqrt{46}}{23}, \frac{5\sqrt{46}}{46}\bigr\rbrack\bigr\rbrack$

About

Examples	7
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