# Understand reflections in the plane

A plane is spanned by the vectors $$\vec{a}$$ and $$\vec{b}$$. The angle between the mirror axes $$\vec{a^\perp}$$ and $$\vec{b^\perp}$$ is equal to the angle of the two vectors $$\vec{a}$$ and $$\vec{b}$$.

Let's say a vector $$\vec{u}$$ is broken down into its components: $$\vec{u} = a \cdot \vec{x} + b \cdot \vec{x^\perp}$$

$$R_x$$ then does the following: $$R_x (a \cdot \vec{x} + b \cdot \vec{x^\perp}) = - a \cdot \vec{x} + b \cdot \vec{x^\perp}$$

With this knowledge, I now try to deduce how this reflection in the graph comes about: Unfortunately, I do not quite understand that. So my ideas are the following: $$R_b \vec{c}$$ would then negate the $$\vec{b}$$ component and $$R_a (R_b \vec{c})$$ would then negate the $$\vec{a}$$ component of $$R_b \vec{c}$$. Then my vector would not be as in the picture above left but bottom left.

So my question is, how does this reflection $$R_a (R_b \vec{c})$$ (picture green) come about? I only have the picture and can not really explain it to me. And if $$R_a (R_b \vec{c})$$ works as in the picture, how does $$-R_a (R_b \vec{v})$$ work? And so that I understand correctly, where can you find exactly the mirror axes? Are these orthogonal on $$\vec{a}$$ and $$\vec{b}$$?

• It would be helpful if you quoted the original source that your question is about. The diagram uses BRA-KET notation, while you aren’t, so something might have gotten lost in translation. Generally speaking, though, the composition of two reflections is a rotation through twice the angle between the mirror lines, so the illustration is misleading. The resulting vector should indeed be pointing down and left. – amd Mar 16 at 20:12
• I agree with your last sentence, if that refers to $R_aR_b|c\rangle$? – QuantaMag Mar 18 at 16:55