What is the Vector Identity for \vec A and \hat n?

[Reshma]

Let \vec A be an arbitrary vector and let \hat n be a unit vector in some fixed direction. Show that
\vec A = (\vec A .\hat n)\hat n + (\hat n \times \vec A)\times \hat n.

 

[TD]

You can use the Vector Triple Product Identity on the last term:

\left( {\vec A \times \vec B} \right) \times \vec C = - \vec A\left( {\vec B \cdot \vec C} \right) + \vec B\left( {\vec A \cdot \vec C} \right)

 

[Maxos]

The vector \vec A is expressed as the sum of its projections on W= \mathcal{L} (\hat{n}) and W^\bot.

Prove that the two terms represent these.