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Rotation Transformation, Rotation Matrix 📂Matrix Algebra

Rotation Transformation, Rotation Matrix

Definition

In the two-dimensional plane R2\mathbb{R}^{2}, the transformation that rotates an arbitrary vector counterclockwise by θ\theta is given by

[x′y′]=[cos⁡θ−sin⁡θsin⁡θcos⁡θ][xy] \begin{bmatrix} x^{\prime} \\ y^{\prime} \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}

Explanation

The matrix [cos⁡θ−sin⁡θsin⁡θcos⁡θ]\begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} is called the rotation matrix or the rotation transformation.

Derivation

Let x=rcos⁡ϕx = r \cos \phi and y=rsin⁡ϕy = r \sin \phi. Let (x′,y′)(x^{\prime}, y^{\prime}) denote the point obtained by rotating the point (x,y)(x, y) by θ\theta. By the trigonometric addition formulas (link), x′,y′x^{\prime}, y^{\prime} are respectively given by

x′=rcos⁡(ϕ+θ)=rcos⁡ϕcos⁡θ−rsin⁡ϕsin⁡θ=xcos⁡θ−ysin⁡θ \begin{align*} x^{\prime} &= r \cos(\phi + \theta) \\ &= r\cos\phi \cos\theta - r\sin\phi \sin\theta \\ &= x \cos\theta - y \sin\theta \\ \end{align*}

y′=rsin⁡(ϕ+θ)=rsin⁡ϕcos⁡θ+rcos⁡ϕsin⁡θ=ycos⁡θ+xsin⁡θ=xsin⁡θ+ycos⁡θ \begin{align*} y^{\prime} &= r \sin(\phi + \theta) \\ &= r\sin\phi \cos\theta + r\cos\phi \sin\theta \\ &= y \cos\theta + x \sin\theta \\ &= x \sin\theta + y \cos\theta \end{align*}

  ⟹  {x′=xcos⁡θ−ysin⁡θy′=ycos⁡θ+xsin⁡θ \implies \begin{cases} x^{\prime} = x \cos\theta - y \sin\theta \\ y^{\prime} = y \cos\theta + x \sin\theta \end{cases}

Writing this system of equations as a matrix (link) yields

[x′y′]=[cos⁡θ−sin⁡θsin⁡θcos⁡θ][xy] \begin{bmatrix} x^{\prime} \\ y^{\prime} \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}

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