In a vector Space normalized, a unit vector is Vecteur whose standard is equal to 1.

This type of vector is used to characterize the direction of an unspecified vector. Thus, one can express a vector $\backslash \; scriptstyle\; \backslash \; vec\; V$ according to an unit vector $\backslash \; vec\; u$ by the Multiplication by a scalar of $\backslash \; vec\; u$ and standard of $\backslash \; scriptstyle\; \backslash \; vec\; V$ (one “stretches” $\backslash \; vec\; u$ of a factor $\backslash \; scriptstyle\; \backslash |\backslash \; vec\; V\; \backslash |$):

$\backslash \; vec\; V\; =\; \backslash |\backslash \; vec\; V\; \backslash |\; \backslash cdot\; \backslash vec\; u$.

To return a vector $\backslash \; unit\; scriptstyle\; \backslash \; vec\; V$, one thus multiplies it by the reverse of his standard.

$\backslash \; vec\; U\; =\; \backslash \; frac\; 1\; \{\backslash |\backslash \; vec\; V\; \backslash |\}\backslash cdot\backslash vec\; V$

Indeed:

$\backslash |\backslash \; vec\; U\; \backslash |\; =\; \backslash \; left\; \backslash |\; \backslash \; frac\; \{1\}\; \{\backslash |\backslash \; vec\; V\; \backslash |\}\; \backslash \; cdot\; \backslash \; vec\; V\; \backslash \; right\; \backslash |\; =\; \backslash \; frac\; \{1\}\; \{\backslash |\backslash \; vec\; V\; \backslash |\}\; \backslash \; cdot\; \backslash |\backslash \; vec\; V\; \backslash |\; =\; 1$

In physics, to indicate the unit vectors, it is usual to use a Circumflex accent: : $\backslash \; hat\; \{X\},\; \backslash \; hat\; \{there\},\; \backslash \; hat\; \{Z\}$.

Derivation of the unit vectors

That is to say a derivable function $t\; \backslash \; mapsto\; E\; (T)$ with values in a Euclidean Space E , such as for all T , E (T) is an unit vector. Then the derived vector e' (T) is Orthogonal with E (T) . Indeed, it is enough to derive the expression from the square to the standard, knowing that this one is constant - thus of null derivative - and that the scalar Produit precisely cancels for two orthogonal vectors:

$\backslash \; frac\; \{\backslash \; mathrm\; D\}\; \{\backslash \; mathrm\; dt\}\; \backslash |E\; (T)\; \backslash |^2\; =\; \backslash \; frac\; \{\backslash \; mathrm\; D\}\; \{\backslash \; mathrm\; dt\}\; \backslash \; langle\; E\; (T)\; \backslash \; mid\; E\; (T)\; \backslash \; rangle\; =\; 2\; \backslash \; langle\; e\text{'}\; (T)\; \backslash \; mid\; E\; (T)\; \backslash \; rangle=0$.

It is the case in particular for the vectors of all the mobile orthonormal bases.