= Cartesian basis under rotation
{c}
{title2=$[p]_{\mathcal U}=Q^Tp$}
Rotate the <standard basis> through $30^\circ$ using the <rotation matrix> whose columns are the new <basis vectors>:
$$
u=\begin{pmatrix}\sqrt3/2\\1/2\end{pmatrix},\quad v=\begin{pmatrix}-1/2\\\sqrt3/2\end{pmatrix},\quad Q=[\,u\ v\,]=\begin{pmatrix}\sqrt3/2&-1/2\\1/2&\sqrt3/2\end{pmatrix},\quad \mathcal U=(u,v).
$$
Here $u\cdot v=0$ and $u\cdot u=v\cdot v=1$, so $Q$ is an <orthogonal matrix> with $Q^TQ=I$, where $T$ denotes <transpose> and $I$ is the <identity matrix>. For the same <vector> $p=(2,1)^T$, its <coordinate vector> in this <Cartesian basis> is
$$
[p]_{\mathcal U}=Q^Tp=\begin{pmatrix}p\cdot u\\p\cdot v\end{pmatrix}=\begin{pmatrix}\sqrt3+1/2\\-1+\sqrt3/2\end{pmatrix}=\begin{pmatrix}\alpha\\\beta\end{pmatrix}.
$$
Thus $\alpha\approx2.232$ and $\beta\approx-0.134$: travel forwards along $u$ and a little backwards along $v$ to reconstruct $p=\alpha u+\beta v$. The <orthogonal projections> onto the two axes give these signed <coordinates> because the measuring <vectors> are both perpendicular and of unit length.
\Image[/cartesian-rotated-orthonormal-coordinates.png]
{title=The <vector> $p=(2,1)^T$ reconstructed from its <orthogonal projections> onto a rotated <Cartesian basis>}
The squared <norm> remains
$$
\alpha^2+\beta^2=(\sqrt3+1/2)^2+(-1+\sqrt3/2)^2=(13/4+\sqrt3)+(7/4-\sqrt3)=5=2^2+1^2.
$$
This is useful when choosing axes aligned with a geometric feature: the <coordinates> change but distances and <dot products> retain the same sum-of-products formula. The two directions of the <change of basis> must be kept straight: $Q$ reconstructs the <vector> in old <coordinates>, $p=Q[p]_{\mathcal U}$, and $Q^T$ extracts its new <coordinates>. Applying $Q$ to the old <coordinate vector> $p$ instead would actively rotate the <vector>.
A short proof explains both formulas. If $p=Qc$, multiplication by $Q^T$ gives $c=Q^Tp$. Then $\|p\|^2=c^TQ^TQc=c^Tc$. Perpendicular unit-length columns make the <matrix inverse> equal the <transpose>; this is the computational convenience supplied by a <Cartesian basis>.
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