Question

Show that in 2 dimensions (say the x , y plane), an inversion through the origin (that is $\mathit{x}\mathbf{\text{'}}\mathbf{=}\mathbf{-}\mathit{x}\mathbf{,}\mathbf{}\mathit{y}\mathbf{\text{'}}\mathbf{=}\mathbf{-}\mathit{y}$) is equivalent to a${\mathbf{180}}^{\mathbf{°}}$rotation of the plane about the axis. Hint:Compare Chapter 3, equation (7.13) with the negative unit matrix.

Short answer

The expressions are proved below.

Step-by-step solution

Given information.

Matrix definitions are given.

Definition of a rotation matrix.

The rotation matrix is defined in this way.

$\left[\begin{array}{cc}\cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \end{array}\right]$

Begin with the definition of matrix inversion.

Define an orthogonal matrix depicting inversion.

A = - l

det A = - 1

This makes it an improper rotation.

Make conclusions from the improper rotation.

Define two vectors, U , V . Since A constitutes an improper rotation, write its consequence on the two vectors.

$$\begin{gathered} \mathrm{U}\text{'}=-\mathrm{U} \\ \mathrm{V}\text{'}=-\mathrm{V} \\ \mathrm{U}\text{'}\times \mathrm{v}\text{'}=(-\mathrm{U})\times (-\mathrm{V}) \\ \mathrm{U}\text{'}\times \mathrm{V}\text{'}=\mathrm{U}\times \mathrm{V} \end{gathered}$$

Hence the expression is proved.