# Help me prove

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

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

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

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.

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