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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. enter image description here

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

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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.

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  • $\begingroup$ 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 $|P−Q|⋅N$? $\endgroup$ – sweets Apr 15 at 17:13
  • $\begingroup$ Oh wait nevermind I get that part now. Thank so much! $\endgroup$ – sweets Apr 15 at 17:16

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