Question
In $D_{4}$, let $K=\left\{R_{0}, R_{90}, R_{180}, R_{270}\right\}$. Write $H R_{90}$ in the form $x H$, where $x \in K$. Write $D R_{270}$ in the form $x D$, where $x \in K .$ Write $R_{90} V$ in the form $V x$, where $x \in K$.
Step 1
We have $H = R_{180}V$, so $HR_{90} = R_{180}VR_{90}$. Since $R_{180}R_{90} = R_{270}$, we have $HR_{90} = R_{270}V$. Now, we want to write this in the form $xH$, where $x \in K$. We can rewrite $R_{270}V$ as $R_{270}(R_{180}H) = (R_{270}R_{180})H = R_{90}H$. So, Show more…
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