Question Number 25783 by lizan 123 last updated on 14/Dec/17
$${If}\:\:\:{r}\:\:{is}\:\:{a}\:\:{unit}\:\:{vector}\:{then}\:\:{show}\:{that}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mid{r}×\frac{{dr}}{{dt}}\mid\:\:=\:\:\mid\frac{{dr}}{{dt}}\mid \\ $$
Answered by ajfour last updated on 15/Dec/17
![r^ =cos θi^ +sin θj^ (dr^ /dt)=((dθ/dt))(−sin θi^ +cos θj^ ) ⇒ ∣(dr^ /dt)∣=absolute value of ( (dθ/dt)) .....(a) while r^ ×(dr^ /dt)=(cos θi^ +sin θj^ )× ((dθ/dt))(−sin θi^ +cos θj^ ) =((dθ/dt))(cos^2 θ+sin^2 θ)k^ =((dθ/dt))k^ ⇒ ∣r^ ×(dr^ /dt)∣=absolute value of ((dθ/dt)) = ∣(dr^ /dt)∣ [see (a) ].](Q25826.png)
$$\bar {{r}}=\mathrm{cos}\:\theta\hat {{i}}+\mathrm{sin}\:\theta\hat {{j}} \\ $$$$\frac{{d}\bar {{r}}}{{dt}}=\left(\frac{{d}\theta}{{dt}}\right)\left(−\mathrm{sin}\:\theta\hat {{i}}+\mathrm{cos}\:\theta\hat {{j}}\right) \\ $$$$\Rightarrow\:\mid\frac{{d}\bar {{r}}}{{dt}}\mid={absolute}\:{value}\:{of}\:\left(\:\frac{{d}\theta}{{dt}}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.....\left(\boldsymbol{{a}}\right)\: \\ $$$${while}\:\bar {{r}}×\frac{{d}\bar {{r}}}{{dt}}=\left(\mathrm{cos}\:\theta\hat {{i}}+\mathrm{sin}\:\theta\hat {{j}}\right)× \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(\frac{{d}\theta}{{dt}}\right)\left(−\mathrm{sin}\:\theta\hat {{i}}+\mathrm{cos}\:\theta\hat {{j}}\right) \\ $$$$\:\:\:\:=\left(\frac{{d}\theta}{{dt}}\right)\left(\mathrm{cos}\:^{\mathrm{2}} \theta+\mathrm{sin}\:^{\mathrm{2}} \theta\right)\hat {{k}} \\ $$$$\:\:\:=\left(\frac{{d}\theta}{{dt}}\right)\hat {{k}} \\ $$$$\Rightarrow\:\:\mid\bar {{r}}×\frac{{d}\bar {{r}}}{{dt}}\mid={absolute}\:{value}\:{of}\:\left(\frac{{d}\theta}{{dt}}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mid\frac{{d}\bar {{r}}}{{dt}}\mid\:\:\:\:\:\:\left[{see}\:\left(\boldsymbol{{a}}\right)\:\right]. \\ $$