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Thread: Proof that a parametrized curve orthogonal to a certain vector v

  1. #1
    Senior Member x3bnm's Avatar
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    Proof that a parametrized curve orthogonal to a certain vector v

    Let $\displaystyle \alpha\colon I\rightarrow R^3$ be a parametrized curve and let $\displaystyle \mathbf{v} \in R^3$ be a fixed vector. Assume that $\displaystyle \alpha'(t)$ is orthogonal to $\displaystyle \mathbf{v}$
    for all $\displaystyle t \in I$ and that $\displaystyle \alpha(0)$ is also orthogonal to $\displaystyle \mathbf{v}$

    Prove that $\displaystyle \alpha(t)$ is orthogonal to $\displaystyle \mathbf{v}$ for all $\displaystyle t \in I$




    We are given that $\displaystyle \alpha'(t)$ and $\displaystyle \alpha(0)$ is orthogonal to $\displaystyle \mathbf{v}$

    Let $\displaystyle \mathbf{v} = (v_1, v_2, v_3)$ and $\displaystyle \alpha(t) = (x(t), y(t), z(t))$

    $\displaystyle \therefore \alpha'(t) = (x'(t), y'(t), z'(t)) $

    So we have:
    $\displaystyle \alpha'(t) \cdot\, \mathbf{v} = (x'(t) * \, v_1 +\,\, y'(t) * v_2 + \,\, z'(t) * v_3) = 0 \text{ [Given]}$

    $\displaystyle \alpha(0) \cdot\, \mathbf{v} = (x(0) *\, v_1 +\,\, y(0) * v_2 +\,\, z(0) * v_3) = 0 \text{ [Given]}$


    Now how do I use this info and prove that $\displaystyle \alpha(t)$ is orthogonal to $\displaystyle \mathbf{v}$ for all $\displaystyle t \in I$?

    Is it possible for anyone to kindly give some hints on how to prove this?
    Last edited by x3bnm; Apr 19th 2012 at 09:21 AM.
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  2. #2
    Senior Member x3bnm's Avatar
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    Re: Proof that a parametrized curve orthogonal to a certain vector v

    I think I solved it. For those who are interested:

    Since

    $\displaystyle \alpha'(t) \cdot\, \mathbf{v} = (x'(t) *\, v_1)+\,\,( y'(t) * v_2) +\,\, (z'(t) * v_3) = 0 \text{ [Given]}$


    Particularly:

    $\displaystyle (x'(t) *\, v_1)+\,\, (y'(t) * v_2) +\,\, (z'(t) * v_3) = 0$


    Taking integration of both sides:

    $\displaystyle (v_1 * x(t)) + C_1 +\,\,( v_2 * (y(t)) + C_2 +\,\, (v_3 * z(t)) + C_3 = C$


    By taking all constants to the right side we get:

    $\displaystyle (v_1 * x(t)) + (v_2 * y(t)) + (v_3 * z(t)) = C - C_1 - C_2 - C_3 \text{ ................(1)}$



    We also know that:

    $\displaystyle \alpha(0) \cdot\, \mathbf{v} = (x(0) *\, v_1) +\,\, (y(0) * v_2) +\,\, (z(0) * v_3) = 0 \text{ [Given]}$


    So pluging in $\displaystyle t = 0$ into eq(1) we get:

    $\displaystyle C - C_1 - C_2 - C_3 = 0$

    therefore:

    $\displaystyle (v_1 * x(t)) + (v_2 * y(t)) + (v_3 * z(t)) = 0$

    But this means: $\displaystyle \alpha(t) \cdot\,\, \mathbf{v} = 0$

    So $\displaystyle \alpha(t)$ is orthogonal to $\displaystyle \mathbf{v}$.[As was to be shown]
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  3. #3
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    Re: Proof that a parametrized curve orthogonal to a certain vector v

    Looks good but I think I would go the "other way"- that is, differentiate $\displaystyle \alpha$ rather than integrate $\displaystyle \alpha'$.

    Let $\displaystyle u(t)= \alpha(t)\cdot v$. Then, since v is a constant vector, $\displaystyle u'= \alpha\cdot v= 0$. Since the derivative of u is 0, u is a constant. And since $\displaystyle u(0)= \alpha(0)\cdot v= 0$ it follows that $\displaystyle u(t)= \alpha(t)\cdot v= 0$ for all t.
    Thanks from x3bnm and skrzatus
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  4. #4
    Senior Member x3bnm's Avatar
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    Re: Proof that a parametrized curve orthogonal to a certain vector v

    Quote Originally Posted by HallsofIvy View Post
    Looks good but I think I would go the "other way"- that is, differentiate $\displaystyle \alpha$ rather than integrate $\displaystyle \alpha'$.

    Let $\displaystyle u(t)= \alpha(t)\cdot v$. Then, since v is a constant vector, $\displaystyle u'= \alpha\cdot v= 0$. Since the derivative of u is 0, u is a constant. And since $\displaystyle u(0)= \alpha(0)\cdot v= 0$ it follows that $\displaystyle u(t)= \alpha(t)\cdot v= 0$ for all t.

    Thank you for showing another way of solving this problem. I believe your way is the correct and better method. Again thanks.
    Last edited by x3bnm; Apr 19th 2012 at 04:40 PM.
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