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Thread: Arbitrary vector

  1. #1
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    Arbitrary vector

    Let $\displaystyle \bold{A} $ be an arbitrary vector and let $\displaystyle \bold{\hat{n}} $ be a unit vector in some fixed direction. Show that $\displaystyle \bold{A} = (\bold{A} \cdot \bold{\hat{n}})\bold{\hat{n}} + (\bold{\hat{n}} \times \bold{A}) \times \bold{\hat{n}} $.

    For this type of problem would I just multiply everything out in the RHS and simplify?

    So $\displaystyle \bold{A} = A \bold{\hat{n}} + 0 $.

    So $\displaystyle (\bold{\hat{n}} \times \bold{A}) \times \bold{\hat{n}} = 0 $?
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  2. #2
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    Here is a basic fact: $\displaystyle A \times \left( {B \times C} \right) = \left( {A \cdot C} \right)B - \left( {A \cdot B} \right)C$.
    In other words, $\displaystyle A \times \left( {B \times C} \right) $ is a linear combination of B & C.

    Now apply that with this:
    $\displaystyle \begin{array}{rcl}
    \left( {N \times A} \right) \times N & = & - \left[ {N \times \left( {N \times A} \right)} \right] \\
    & = & - \left[ {\left( {N \cdot A} \right)N - \left( {N \cdot N} \right)A} \right] \\
    & = & A - \left( {N \cdot A} \right)N \\ \end{array}
    $

    Can you see how to finish?
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