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Math Help - Vector Proof - Dot and Cross Products

  1. #1
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    Vector Proof - Dot and Cross Products

    Hello everyone,

    I am having some trouble with the following problem and would appreciate help. Thank you!

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    16. For any vectors  \vec{a}, \vec{b}, \vec{c} , show that  (\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \bullet \vec{c})\vec{b} - (\vec{b} \bullet \vec{c})\vec{a} .

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    I worked on the RS since it is more complicated:

    RS =  (|\vec{a}||\vec{c}|\cos \theta)\vec{b} - (|\vec{b}||\vec{c}|\cos \theta)\vec{a})

    However, I do not know how to distribute the vectors  \vec{b} and  \vec{a} into the dot products.
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  2. #2
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    Have you already proven this <br />
a \times (b \times c) = \left( {a \cdot c} \right)b - \left( {a \cdot b} \right)c?
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  3. #3
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    Quote Originally Posted by Plato View Post
    Have you already proven this <br />
a \times (b \times c) = \left( {a \cdot c} \right)b - \left( {a \cdot b} \right)c?

    Thanks for your response, Plato.

    I have not proven what you wrote but I do understand it; <br />
a \times (b \times c) =  (a \times b) \times c because the vector perpendicular to these vectors would be equal regardless of the order of the calculations.

    How would I use your identity to simplifying the RS?
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  4. #4
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    Actually a \times (b \times c) \ne \left( {a \times b} \right) \times c.
    Because the LHS is a linear combination of b\;\&\; c.
    The RHS is a linear combination of a\;\&\; b.

    So as painful as it is the best way is to use coordinates for each vector.
    It is a messy, messy exercise in subscripts and factoring.
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  5. #5
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    Quote Originally Posted by Plato View Post
    Actually a \times (b \times c) \ne \left( {a \times b} \right) \times c.
    Because the LHS is a linear combination of b\;\&\; c.
    The RHS is a linear combination of a\;\&\; b.

    So as painful as it is the best way is to use coordinates for each vector.
    It is a messy, messy exercise in subscripts and factoring.
    Thank you for your reply, Plato.

    Would you mind elaborating on a \times (b \times c) \ne \left( {a \times b} \right) \times c?
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