A 2×2 orthogonal matrix
The matrix
![\[U = \dfrac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\]](https://ecampusontario.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-e86029b5c19592fd738be83f558869b6_l3.png "Rendered by QuickLaTeX.com")
is orthogonal.
The vector 
is transformed by the orthogonal matrix above into
![\[Ux = \dfrac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 3 \end{pmatrix}.\]](https://ecampusontario.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-bd5e01ac2ff0e3bb1ddb29dee842b2f1_l3.png "Rendered by QuickLaTeX.com")
Thus, 
corresponds to a rotation of angle 
degrees counter-clockwise.
The matrix
is orthogonal.
The vector is transformed by the orthogonal matrix above into
Thus, corresponds to a rotation of angle degrees counter-clockwise.