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 $?