derivation of rotation matrix using polar coordinates

We derive formally the expression for the rotation of a two-dimensional vector$𝒗=a𝒙+b𝒚$ by an angle $\varphi$ counter-clockwise. Here$𝒙$ and $𝒚$ are perpendicular unit vectors that are oriented counter-clockwise(the usual orientation).

In terms of polar coordinates, $𝒗$ may be rewritten:

$𝒗$$=r(\cos \theta 𝒙+\sin \theta 𝒚),a=r\cos \theta ;b=r\sin \theta ,$

for some angle $\theta$ and radius $r\ge 0$.To rotate a vector $𝒗$ by $\varphi$ really means to shift itspolar angle by a constant amount $\varphi$ but leave its polar radius fixed.Therefore, the result of the rotation must be:

${𝒗}^{\prime }$$=r\left(\cos (\theta +\varphi )𝒙+\sin (\theta +\varphi )𝒚\right)$

Expanding using the angle addition formulae, we obtain

${𝒗}^{\prime }$$=r(\cos \theta \cos \varphi -\sin \theta \sin \varphi )𝒙+(\sin \theta \cos \varphi +\cos \theta \sin \varphi )𝒚)$

$=(a\cos \varphi -b\sin \varphi )𝒙+(b\cos \varphi +a\sin \varphi )𝒚.$

When this transformation is written out in $[𝒙,𝒚]$-coordinates, we obtain the formula for the rotation matrix: