What is the geometric meaning of the transformation of $\mathbb{R}^2$ when every vector is multiplied by -1?

Question

Is it a rotation?

I think it is. So for example if we multiply by U$ = (U_1,U_2)$ by $-1$,

We'll have $-U_1, -U_2$. Then the vector is heading on the opposite direction.

Is it enough to just say, yes it is because multiplying by $-1$ will lead the vectors to the opposite direction of rotation by $180$ degrees?

Answer 1

The general rotation matrix for $\mathbb{R}^2$ is given by,

$$ A=\begin{pmatrix} \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$$

Can you give a $\theta$ such that $A \textbf{x} = - \textbf{x}$ where $\textbf{x} = (x,y)$?

Answer 2

Yes, it is a rotation. In general, if $\theta$ is a given angle, $$\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \cos \theta - y \sin \theta \\ x \sin \theta + y \cos \theta\end{pmatrix}$$is the point obtained by rotating $(x,y)$ by $\theta$, counter-clockwise. Plug in $\theta = \pi$ and see the magic happen.

Answer 3

Yes, it is enough to say "because multiplying by −1 will lead the vectors to the opposite direction of rotation by 180 degrees." In the complex plane the factor $e^{i\theta}$ is an anticlockwise rotation of $\theta$ and $e^{i\pi}=-1$.