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Math Help - Derivative of a vector function

  1. #1
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    Derivative of a vector function

    I'm having trouble with this problem I was wondering if someone could help me out.

    If r(t) is not equal to zero, show that
    (d/dt)|r(t)| = 1/|r(t)|r(t)*r'(t)
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  2. #2
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    Quote Originally Posted by Todeezy View Post
    I'm having trouble with this problem I was wondering if someone could help me out.

    If r(t) is not equal to zero, show that
    (d/dt)|r(t)| = 1/|r(t)|r(t)*r'(t)

    r(t)=<x(t),y(t),z(t)>

    r'(t)=<\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt}>

    \mid r(t) \mid =\sqrt{x^2+y^2+z^2}

    \frac{d}{dt}\mid r(t) \mid =\frac{1}{2\sqrt{x^2+y^2+z^2}}\frac{d}{dt}(x^2+y^2  +z^2)

    Notice that the derivative of the inner function is just the dot product of the vector and its derivative.

    \frac{d}{dt}(x^2+y^2+z^2)=2x\frac{dx}{dt}+2y\frac{  dy}{dt}+2z\frac{dz}{dt}=2(r(t)\cdot r'(t))

    \frac{d}{dt}\mid r(t) \mid =\frac{2(r(t)\cdot r'(t))}{2\sqrt{x^2+y^2+z^2}}

    =\frac{r(t)\cdot r'(t)}{\mid r(t) \mid}
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