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Question Number 25783 by lizan 123 last updated on 14/Dec/17

$$Ifrisaunitvectorthenshowthat$$$$\mid r\times \frac{dr}{dt}\mid =\mid \frac{dr}{dt}\mid$$

Answered by ajfour last updated on 15/Dec/17

$$\overline{r}=\cos \theta \hat{i}+\sin \theta \hat{j}$$$$\frac{d\overline{r}}{dt}=\left(\frac{d\theta }{dt}\right)(-\sin \theta \hat{i}+\cos \theta \hat{j})$$$$\Rightarrow \mid \frac{d\overline{r}}{dt}\mid =absolutevalueof\left(\frac{d\theta }{dt}\right)$$$$.....\left(\mathit{a}\right)$$$$while\overline{r}\times \frac{d\overline{r}}{dt}=(\cos \theta \hat{i}+\sin \theta \hat{j})\times$$$$\left(\frac{d\theta }{dt}\right)(-\sin \theta \hat{i}+\cos \theta \hat{j})$$$$=\left(\frac{d\theta }{dt}\right)(\cos {}^2\theta +\sin {}^2\theta )\hat{k}$$$$=\left(\frac{d\theta }{dt}\right)\hat{k}$$$$\Rightarrow \mid \overline{r}\times \frac{d\overline{r}}{dt}\mid =absolutevalueof\left(\frac{d\theta }{dt}\right)$$$$=\mid \frac{d\overline{r}}{dt}\mid \left[see\left(\mathit{a}\right)\right].$$

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