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Math Help - Arbitrary vector

  1. #1
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    Arbitrary vector

    Let  \bold{A} be an arbitrary vector and let  \bold{\hat{n}} be a unit vector in some fixed direction. Show that  \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  \bold{A} = A \bold{\hat{n}} + 0 .

    So  (\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: A \times \left( {B \times C} \right) = \left( {A \cdot C} \right)B - \left( {A \cdot B} \right)C.
    In other words, A \times \left( {B \times C} \right) is a linear combination of B & C.

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

    Can you see how to finish?
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