= Solution
The displayed <orthogonal matrix> transforms the coordinates of a fixed <vector>, rather than actively moving that vector. Write $\theta=\arctan(1/2)$. Then $P=R(-\theta)$, so the new unit basis vectors, expressed in the old <orthonormal basis>, are the columns of $P^T$:
$$
\boxed{\mathbf e'_1=\frac{(2,1)}{\sqrt5},\qquad
\mathbf e'_2=\frac{(-1,2)}{\sqrt5}.}
$$
The axes have rotated counterclockwise through $\theta$, while the coordinate column transforms by the opposite rotation. For the example point $\mathbf r=(1,0)$, its new <coordinates> are $(2,-1)/\sqrt5$. Reconstruction gives $(2\mathbf e'_1-\mathbf e'_2)/\sqrt5=(1,0)$, confirming that the point is unchanged. The middle panel of the figure in part <1> shows these axes and the signed projections. Since $P^TP=I$ and $\det P=1$, lengths and orientation of the basis are preserved.
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