Expected value of the inner product of two random vectors $X=[x_1, x_2,...x_n] Y = [y_1, y_2,...y_n]$
If $x_i, y_i$ are both random variables with 
$P(x=1) = .5$
$ P(x=2) = .5 $
$P(y=1)=.5$
$P(y=2)=.5$
How would I find the expected value of the inner product of both of these random vectors?$E[X.Y]$
I'm thrown off with this problem, as I don't know how exactly to work with random vectors or if any additional rules apply when finding the expected value.
My gut feeling is
$E[X.Y] = [E[x_1y_1], E[x_2y_2],.....,E[x_ny_n]]$
Is this the right approach?

$$E[X.Y] = E[\sum_{1}^{n} (x_{i}y_{i})]=\sum_{1}^{n} E[x_{i}y_{i}]=\sum_{1}^{n}2.25=2.25n$$
Explanation:
1) Definition of dot product of two vectors where $X=[x_{1},x_{2},...x_{n}]$ and $Y=[y_{1},y_{2},...y_{n}]$: $X.Y = \sum_{1}^{n} (x_{i}y_{i}) $ so you have: $E[X.Y] = E[\sum_{1}^{n} (x_{i}y_{i})]$
2) $E[a+b] = E[a]+E[b]$ so $E[\sum_{1}^{n} (x_{i}y_{i})]=\sum_{1}^{n}E[x_{i}y_{i}]$
3) Say $z_{i} = x_{i}y_{i}$ then the pdf of $z_{i}$ is given by (let me know if you need further explanation here): $$z_{i} = \left\{\begin{matrix}
1 & .25 \\
2 & .5\\ 
4 & .25
\end{matrix}\right.$$
So using the definition of expectation we get: $$E[z_{i}]=\sum_{1}^{3}z_{i}=2.25=E[x_{i}y_{i}]$$
4)$$\sum_{1}^{n}E[x_{i}y_{i}]=\sum_{1}^{n}2.25=2.25n$$
