How can I prove this statement about two vectors?

Question

How can I go about proving that $A=(A\cdot \hat n)\hat n+(\hat n\times A)\times \hat n$ where "$\cdot$" and "$\times$" indicate the dot and cross products respectively, $A$ is an arbitrary vector, and $\hat n$ is a unit vector in some fixed direction?

Answer 1

Use the vector triple product and the fact that $\hat n \cdot \hat n = 1$:

$$A=(A\cdot \hat n)\hat n+(\hat n\times A)\times \hat n$$

$$=(A\cdot \hat n)\hat n+ \left(-(A \cdot \hat n)\hat n + (\hat n\cdot \hat n)A\right)$$

$$=(A\cdot \hat n)\hat n- (A \cdot \hat n)\hat n + A$$

$$= A$$