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Prove Vector Derivative (using Dot Product)

  1. May 16, 2010 #1
    1. The problem statement, all variables and given/known data
    n.

    Let r1 and r2 be differentiable 3-space vector-valued functions.

    Directly from the definition of dot product, and the definition of derivative of vector-
    valued functions in terms of components, prove that
    d/dt (r1(t) • r2(t)) = r′1(t) • r2(t) + r1(t) • r′2(t).

    2. Relevant equations

    Dot Product: U • V = u1v1 + u2v2 + u3v3

    3. The attempt at a solution

    I can just substitute things forever and get nowhere...
     
  2. jcsd
  3. May 17, 2010 #2

    lanedance

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    Homework Helper

    show us how?

    in this example the dot product is a scalar valued function of t, use the component form you have shown & the product rule and it should follow straight forward
     
    Last edited: May 17, 2010
  4. May 17, 2010 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    But have you differentiated?

    Since, as you write, U• V= u1v1+ u2v2+ u3v3, (U• V)'= u1'v1+ u1v1'+ u2'v2+ u2v2'+ u3'v3+ u3v3'= (u1'v1+ u2'v2+ u3'v3)+ (u1v1'+ u2v2'+ u3v3').
     
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