If-r-is-a-unit-vector-then-show-that-r-dr-dt-dr-dt-

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

I f r i s a u n i t v e c t o r t h e n s h o w t h a t

∣ r \times \frac{d r}{d t} ∣ = ∣ \frac{d r}{d t} ∣

Answered by ajfour last updated on 15/Dec/17

\bar{r} = \cos θ \hat{i} + \sin θ \hat{j}

\frac{d \bar{r}}{d t} = (\frac{d θ}{d t}) (- \sin θ \hat{i} + \cos θ \hat{j})

\Rightarrow ∣ \frac{d \bar{r}}{d t} ∣= a b s o l u t e v a l u e o f (\frac{d θ}{d t})

\dots . . (a)

w h i l e \bar{r} \times \frac{d \bar{r}}{d t} = (\cos θ \hat{i} + \sin θ \hat{j}) \times

(\frac{d θ}{d t}) (- \sin θ \hat{i} + \cos θ \hat{j})

= (\frac{d θ}{d t}) (\cos^{2} θ + \sin^{2} θ) \hat{k}

= (\frac{d θ}{d t}) \hat{k}

\Rightarrow ∣ \bar{r} \times \frac{d \bar{r}}{d t} ∣= a b s o l u t e v a l u e o f (\frac{d θ}{d t})

= ∣ \frac{d \bar{r}}{d t} ∣ [s e e (a)] .

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