# Math Help - Arbitrary vector

1. ## 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 $(\bold{\hat{n}} \times \bold{A}) \times \bold{\hat{n}} = 0$?

2. 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}
\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?