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Math Help - Why's that normal = binormal x tangent for a curve parametrized by arc length s?

  1. #1
    Senior Member x3bnm's Avatar
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    Why's that normal = binormal x tangent for a curve parametrized by arc length s?

    I'm stuck at the properties of a 3d curve parametrized by arc length. My problem is given below.


    We know that the binormal of a curve at a point is:

    b = n \times t

    where b = binormal and n is normal to the curve and t is tangent.

    So binormal is the cross product of normal and tangent at a point of a curve.



    Now if b = n \times t why's that n = b \times t ? How do one prove this statement?

    What properties of mathematics verify that n = b \times t? Is it possible to kindly help me answer this question?
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  2. #2
    MHF Contributor

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    Re: Why's that normal = binormal x tangent for a curve parametrized by arc length s?

    With b= n\times t, there is NO "property of mathematics" that gives " n= b\times t". It is, rather, true that n= t\times b.

    n and t are also unit vectors. Further, the length of u\times v is |u||v|sin(\theta) where \theta is the angle between the two vectors. Since n and t are at right angles, b is also a unit vector, perpendicular to both n and t. That is, we have the "cyclic" formulas, b= n\times t, n= t\times b and t= b\times n.
    Thanks from x3bnm
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    Senior Member x3bnm's Avatar
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    Re: Why's that normal = binormal x tangent for a curve parametrized by arc length s?

    Thanks HallsofIvy.

    Then if a = b \times c for any vector a, b and c then can we say b = a \times c and c = a \times b?
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    Senior Member x3bnm's Avatar
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    Re: Why's that normal = binormal x tangent for a curve parametrized by arc length s?

    x3bnm said:
    Then if a = b \times c for any vector a, b and c then can we say b = a \times c \text{ and } c = a \times b?

    I got it. My last statement is wrong. For those who want to understand: the Counterexample to my statement:

    Suppose b = (1,2,1), c = (2, 3, 1) \text{ and } a = b \times c = (-1, 1, -1)

    But b \neq a \times c \text{ and } c \neq a \times b


    The statement that n = b \times t is true for binormal, normal and tangent.
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