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Thread: Local Coords

  1. #1
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    Local Coords

    If $\displaystyle \bold{r}(t) = (t^2-t, t \sqrt{2t-t^2}), 0 \leq t \leq 2 $ is $\displaystyle \bold{u}_r = \frac{(t^2-t, t \sqrt{2t-t^2})}{\sqrt{(t^2-t)^{2} + t^{2}(2t-t^2)}} $ and $\displaystyle \bold{u}_\theta = \frac{(t \sqrt{2t-t^2}, t^2-t)}{\sqrt{(t^2-t)^{2} + t^{2}(2t-t^2)}} $?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by shilz222 View Post
    If $\displaystyle \bold{r}(t) = (t^2-t, t \sqrt{2t-t^2}), 0 \leq t \leq 2 $ is $\displaystyle \bold{u}_r = \frac{(t^2-t, t \sqrt{2t-t^2})}{\sqrt{(t^2-t)^{2} + t^{2}(2t-t^2)}} $ and $\displaystyle \bold{u}_\theta = \frac{(t \sqrt{2t-t^2}, t^2-t)}{\sqrt{(t^2-t)^{2} + t^{2}(2t-t^2)}} $?
    what are $\displaystyle \bold{u}_r \mbox { and } \bold{u}_{\theta}$ ?

    they seem to be unit vectors, but unit vectors of what? or in what direction?
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    $\displaystyle \bold{u}_r = \frac{\bold{r}(t)}{r(t)} $ in the direction of $\displaystyle \bold{r} $ and $\displaystyle \bold{u}_\theta $ is a perpendicular unit vector.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by shilz222 View Post
    $\displaystyle \bold{u}_r = \frac{\bold{r}(t)}{r(t)} $ in the direction of $\displaystyle \bold{r} $ and $\displaystyle \bold{u}_\theta $ is a perpendicular unit vector.
    you mean $\displaystyle \bold{u}_r = \frac {\bold{r}(t)}{|\bold{r}(t)|}$ ?

    note that: for a vector $\displaystyle <a,b>$ two perpendicular vectors are $\displaystyle <-b,a>$ or $\displaystyle <b,-a>$, they point in opposite directions. you put the vector $\displaystyle <b,a>$
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    yes, that was a typo. yeah in the book, $\displaystyle r(t) = ||\bold{r}(t)|| $.
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