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Thread: Vector valued function problem

  1. #1
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    Vector valued function problem

    "Show that, at a local max or min of $\displaystyle ||\vec{r}(t)||$, the vector $\displaystyle r'(t)$ is perpendicular to $\displaystyle \vec{r}(t)$"

    I think that if $\displaystyle f(t) = ||\vec{r}(t)||^2$, then the local min/max of $\displaystyle f(x)$ should be the same as $\displaystyle ||\vec{r}(t)||$, but I am not quite sure how to proceed with the rest of the question after that or how to show that $\displaystyle r'(t)$ is perpendicular to $\displaystyle \vec{r}(t)$.
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  2. #2
    A Plied Mathematician
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    Here are a few relevant ideas:

    1. $\displaystyle \|\mathbf{r}(t)\|^{2}=\mathbf{r}(t)\cdot\mathbf{r} (t)$

    2. If $\displaystyle \|\mathbf{r}(t)\|$ has a max or min, then $\displaystyle \|\mathbf{r}(t)\|^{2}$ does as well. Proof: assume

    $\displaystyle \dfrac{d}{dt}\|\mathbf{r}(t)\|=0.$

    Then $\displaystyle \dfrac{d}{dt}\|\mathbf{r}(t)\|^{2}=2\|\mathbf{r}(t )\|\,\dfrac{d}{dt}\|\mathbf{r}(t)\|=0.$

    QED.

    3. The product rule for derivatives works for the dot product.

    There are some hints for you.
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