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Thread: Vector geometry question

  1. #1
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    Orthogonal vectors

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

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

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

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

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

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

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