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Math Help - Vector geometry question

  1. #1
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    Orthogonal vectors

    Let \mathbf{a,b,c} be three vectors in \mathbb{R}^3 which satisfy the relations \mathbf{b=c \times a}\ and\ \mathbf{c=a \times b}

    1) Show that \mathbf{a, b\ and\ c} are a set of mutually orthogonal vectors

    2) Show that \mathbf{b\ and\ c} are of equal length and that if \mathbf{b \neq 0}, then \mathbf{a} is a unit vector.
    Last edited by acevipa; June 1st 2010 at 06:26 AM.
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  2. #2
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    Quote Originally Posted by acevipa View Post
    Let \mathbf{a,b,c} be three vectors in \mathbb{R}^3 which satisfy the relations \mathbf{b=c \times a}\ and\ \mathbf{c=a \times b}

    1) Show that \mathbf{a, b\ and\ c} are a set of mutually orthogonal vectors

    2) Show that \mathbf{b\ and\ c} are of equal length and that if \mathbf{b \neq 0}, then \mathbf{a} is a unit vector.
    This is pretty straight forward isn't it? \mathbf{b= c\times a} is, by definition of "cross product", orthogonal to both \mathbf{a} and mathbf{c} and \mathbf{c= a\times b} is orthogonal to a.

    Further, since \mathbf{a} is orthogonal to \mathbf{c}, |\mathbf{b}|= |\mathbf{a}\times\mathbf{c}|= |\mathbf{a}||\mathbf{c}\sin(\pi/2) so that

    1= (|\mathbf{a}|)(1)(1).
    Last edited by Isomorphism; June 1st 2010 at 03:07 AM. Reason: Iso:Changed cos to sin to avoid confusion
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