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Vectors: Find w such that dv/ds = w x v

  1. Sep 22, 2012 #1
    1. The problem statement, all variables and given/known data

    T (tangent), B (binormal), N (normal) are an orthogonal triad of unit vectors of a curve in R3.


    Given:
    • dT/ds = kN
    • dN/ds = -kT + tB
    • dB/ds = -tN

    Find vector w so that these equations may be written in the form:

    dv/ds = w x v, where v = T + N + B




    2. Relevant equations

    Given above!




    3. The attempt at a solution

    I tried splitting v (T + N + B) into its components, as well as w, and putting them into a matrix to find the determinant (for the cross product).

    The matrix consists of <i, j, k>, w = <w1, w2, w3>, and v = <(T1 + N1 + B1), (T2 + N2 + B2), (T3 + N3 + B3)>.

    However, taking the determinant/cross product gives mismatching components (ie. j and k components supposedly add up to an i component). I'm not too sure where to go from here :(



    Note: This is for a Multivariate Calculus course!
     
    Last edited: Sep 22, 2012
  2. jcsd
  3. Sep 22, 2012 #2

    gabbagabbahey

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    Hi YayMathYay, welcome to PF! :smile:

    There's a much easier way to do this than by looking at the Cartesian components of the vectors.

    Hint: What is [itex]\mathbf{v}\times ( \mathbf{w} \times \mathbf{v} )[/itex]?
     
  4. Sep 22, 2012 #3
    First of all, thanks for the welcome! :)

    I know the hint you gave me means [itex]\left(\mathbf{v} \bullet \mathbf{v}\right)\mathbf{w} - \left(\mathbf{v} \bullet \mathbf{w}\right)\mathbf{v}[/itex]. I've tried this method before, but I wasn't too sure where to go from here..
     
  5. Sep 22, 2012 #4

    gabbagabbahey

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    For starters, what is [itex]\mathbf{v} \cdot \mathbf{v}[/itex] ? Next since [itex]\mathbf{T}[/itex], [itex]\mathbf{B}[/itex] & [itex]\mathbf{N}[/itex] are an orthogonal triad in [itex]\mathbb{R}^3[/itex], they span [itex]\mathbb{R}^3[/itex], and so any vector can be decomposed into a linear combination of those unit vectors. So, why not write [itex]\mathbf{w} = w_{T}\mathbf{T} + w_{N}\mathbf{N} + w_{B}\mathbf{B}[/itex]... what does that make [itex]\mathbf{w} ( \mathbf{v} \cdot \mathbf{v} ) - \mathbf{v} ( \mathbf{w} \cdot \mathbf{v} )[/itex]?
     
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