The figure from below shows two systems of cartesian coordinates. On ($x,y$) vector $\vec{A}$ is expressed as $\vec{A}=5\hat{i}+4\hat{j}$. Find the vector $\sqrt{2}\vec{A}$ on the system $x',y'$.

The alternatives given are as follows:

$\begin{array}{ll}

1.&9\hat{i'}+\hat{j'}\\

2.&\hat{i'}+9\hat{j'}\\

3.&-9\hat{i'}+\hat{j'}\\

4.&9\hat{i'}-\hat{j'}\\

5.&\hat{i'}-9\hat{j'}\\

\end{array}$

Does it exist a way to solve this problem visually or with least use of algebra? The only thing which I could spot for the vector given is that the angle for the vector is:

$\tan\omega=\frac{4}{5}$

therefore $\omega=\tan^{-1}\left(\frac{4}{5}\right)$

This angle is not very known. How exactly can I relate it with the tilt in the new system of coordinates? What exactly is what should I do?