= Cartesian frame
{c}
{title2=$\mathcal F=(O;e_1,\ldots,e_n)$}
A <Cartesian frame> combines a chosen origin <point> $O$ with a <Cartesian basis> $\mathcal E$, including its ordering as an <ordered basis>. A <point> $P$ has <coordinate vector> $c$ when $\overrightarrow{OP}=\sum_i c_i e_i$. The <basis> supplies measuring directions and unit lengths; the <coordinate vector> is the resulting <tuple> of numbers; the <Cartesian frame> additionally fixes where measurement starts. A <vector> already belongs to a <vector space> with a <zero vector>, so expressing it in a <basis> requires no additional choice of spatial origin.
For example, keeping the <standard basis> but moving the origin from $O=(0,0)$ to $O'=(1,0)$ changes the <coordinates> of the physical <point> $P=(2,1)$ from $(2,1)^T$ to $(1,1)^T$. This happens because $\overrightarrow{O'P}=\overrightarrow{OP}-\overrightarrow{OO'}$ is a different displacement <vector>. By contrast, the two diagrams above keep $O$ fixed and change only the <basis>. In a fixed reference <Cartesian frame>, a new frame with origin column $o$ and basis-column <matrix> $Q$ describes <points> by $p=o+Qc$; the new <coordinates> are $c=Q^T(p-o)$ when $Q$ is an <orthogonal matrix>.
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