Orthonormal base

Either E _N a Euclidean vector space of Dimension N , where N is a Entier naturalness not no one, and $\ mathcal B = (\ vec e_1, \ vec e_2,…, \ vec e_n)$ , a bases E _N.

If N = 1, then $\ mathcal B = (\ vec e_1)$ is known as orthonormal if and only if

\| \ vec e_1 \| = 1

If N > 1, then $\ mathcal B$ is orthonormal if and only if

\| \ vec e_1 \| = \| \ vec e_2 \| =… = \| \ vec e_n \| = 1

and,

for all

I \ not = j

\ vec e_i \ perp \ vec e_j

(i.e.

\ vec e_i \ cdot \ vec e_j

= 0)

The term “bases orthonormée” is sometimes GOOD shortened by the Sigle.

Orthonormal reference mark (or orthonormé)

Is has _n a space refines Euclidean associated with the Euclidean vector space E _N and O an unspecified point of has _n, then the reference mark

\ mathcal R = (\ O, \ vec e_1, \ vec e_2,…, \ vec e_n)

is known as orthonormal if and only if its bases associated

\ mathcal B = (\ vec e_1, \ vec e_2,…, \ vec e_n)

is itself orthonormal.

The term “locates orthonormal” is sometimes shortened by the Sigle RON.

In solid geometry

In Solid geometry, the base is in general noted $(\ vec {I}, \ vec {J}, \ vec {K})$ instead of $(\ vec {e_1}, \ vec {e_2}, \ vec {e_3})$ .

The base is known as “direct” if $\ vec {K}$ is the vector Product of $\ vec {I}$ and of $\ vec {J}$ ( $\ vec {K} = \ vec {I} \ wedge \ vec {J}$ ).

The term “bases orthonormal direct” is sometimes shortened by initials BOD.

If the base associated with a reference mark is orthonormal direct, the reference mark is a direct orthonormal reference mark, term sometimes shortened by the ROUND Sigle.

See the article Orientation (mathematics).

See too

Cartesian Frame of reference

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