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Thread: Vector Product

  1. #1
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    Vector Product

    Dear all,

    I have a vector identity and am confused about the notation

    $\displaystyle
    \bf a\times (\bf b\times \bf c)=\bf b(\bf a\cdot \bf c)-\bf c(\bf a\cdot\bf b)
    $

    where $\displaystyle \bf a$, $\displaystyle \bf b$ and $\displaystyle \bf c$ are vectors and $\displaystyle \times$ and $\displaystyle \cdot$ are the cross and dot product respectively.

    My question is what is the operation between $\displaystyle \bf b(a$?

    It isn't the dot product. Is it the scalar product? Also is it commutative?

    $\displaystyle
    \bf b(\bf a\cdot \bf c)-\bf c(\bf a\cdot\bf b)=(\bf a\cdot \bf c)\bf b-(\bf a\cdot\bf b)\bf c
    $
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  2. #2
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    Quote Originally Posted by davefulton View Post
    Dear all,

    I have a vector identity and am confused about the notation

    $\displaystyle
    \bf a\times (\bf b\times \bf c)=\bf b(\bf a\cdot \bf c)-\bf c(\bf a\cdot\bf b)
    $

    where $\displaystyle \bf a$, $\displaystyle \bf b$ and $\displaystyle \bf c$ are vectors and $\displaystyle \times$ and $\displaystyle \cdot$ are the cross and dot product respectively.

    My question is what is the operation between $\displaystyle \bf b(a$?

    It isn't the dot product. Is it the scalar product? Also is it commutative?

    $\displaystyle
    \bf b(\bf a\cdot \bf c)-\bf c(\bf a\cdot\bf b)=(\bf a\cdot \bf c)\bf b-(\bf a\cdot\bf b)\bf c
    $

    The only possible, and sensical, meaning of $\displaystyle b(a\cdot c)$ is the vector $\displaystyle b$ multiplied by the scalar $\displaystyle a\cdot c$ ...Now you prove this.

    Tonio

    Pd. I'm assuming you're working with vectors in $\displaystyle \mathbb{R}^3$...?
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  3. #3
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    The dot product of any two vectors is a scalar. This answers my question. Thank you. I suppose this implies it is commutative also.
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  4. #4
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    Quote Originally Posted by davefulton View Post
    The dot product of any two vectors is a scalar. This answers my question. Thank you. I suppose this implies it is commutative also.

    Well, the product of a scalar and a vector is obviously commutative, as is the dot product, but NOT the cross product!

    Tonio
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