Suppose P and Q are two distinct points. Prove that there exists exactly one translation which sends P to Q

ganolrifv9
2022-07-30
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yermarvg

Answered 2022-07-31
Author has **19** answers

$\text{Let}T:{R}^{n}\to {R}^{n}\text{be a translation}\phantom{\rule{0ex}{0ex}}\Rightarrow T(x)=x+aforsomea\in {R}^{n}\phantom{\rule{0ex}{0ex}}\text{Now we need}T(P)=Q\phantom{\rule{0ex}{0ex}}i.e.,T(P)=P+a=Q\phantom{\rule{0ex}{0ex}}\Rightarrow a=Q-P\phantom{\rule{0ex}{0ex}}\therefore \text{if a tran slation map wants to send P to Q it has to be the unique translation}\phantom{\rule{0ex}{0ex}}\text{namely}T(x)=x+(Q-P)fora\in {R}^{n}\phantom{\rule{0ex}{0ex}}$

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Sorry if this is just a stupid assumption... it may be one of those things that just seems correct but is actually wrong.