I don't understand the solution to this question:

Find the component of the force $\displaystyle \bold{F}=(3\bold{i}+2\bold{j})N$ in the direction of the vector $\displaystyle 2\bold{i}-\bold{j}$.

Solution:

Let $\displaystyle \bold{a}=2\bold{i}-\bold{j}$. Then the unit vector in the direction of $\displaystyle \bold{a}$ is $\displaystyle \hat{\bold{a}}=\frac{1}{\sqrt{5}}(2\bold{i}-\bold{j})$.

$\displaystyle \bold{F}\cdot \hat{\bold{a}}=(3\bold{i}+2\bold{j})\cdot \frac{1}{\sqrt{5}}(2\bold{i}-\bold{j})=\frac{4\sqrt{5}}{5}$

and $\displaystyle (\bold{F}\cdot \hat{\bold{a}})\cdot \hat{\bold{a}}=\frac{4\sqrt{5}}{5}\times \frac{1}{\sqrt{5}}(2\bold{i}-\bold{j})=\frac{4}{5}(2\bold{i}-\bold{j})$

Hence the component of $\displaystyle \bold{F}$ in the direction of $\displaystyle 2\bold{i}-\bold{j}$ is $\displaystyle \frac{4}{5}(2\bold{i}-\bold{j})N$

What do the expressions $\displaystyle \bold{F}\cdot \hat{\bold{a}}$ and $\displaystyle (\bold{F}\cdot \hat{\bold{a}})\cdot \hat{\bold{a}}$ mean? How in general do you solve these problems? Thanks