# Euclidean Geometry Translations

We haven't covered translations yet but we have problems on it so I'm reading the textbook. From what I understand the composition of two reflections is a translation? This was the explanation in the book.

I understand lines 1-3. My confusion comes from how did they get from 3 to 4. Thank you for your time

Recall the following formulation of the scalar product of two vectors $$v, w$$: $$\langle v , w\rangle = |v| \cdot |w| \cdot \cos(\varphi)$$ where $$\varphi$$ is the angle between $$v$$ and $$w$$. In your situation the vector $$P-Q$$ points in the same (or opposite) direction as $$N$$, because the line $$PQ$$ is perpendicular to $$\mathit{m}$$. Thus you either get $$\cos(\varphi)=1$$ $$\langle P-Q , N\rangle N = (|P-Q|\cdot |N|) \cdot N = |P-Q| \cdot N = P-Q$$ if $$N$$ and $$P-Q$$ point in the same direction, or $$\cos(\varphi)=-1$$ $$\langle P-Q , N\rangle N = (|P-Q|\cdot |N| \cdot (-1) )\cdot N = (-1) \cdot|P-Q| \cdot N = (-1) (Q-P) = P-Q$$ if $$N$$ and $$P-Q$$ point in opposite directions.
• So I drew out the diagram and I see what you mean by they point in the same (or opposite direction) . Can I ask how $(|P−Q|⋅|N|)⋅N$ becomes $|P−Q|⋅N$? – sweets Apr 15 at 17:13