$$\begin{aligned} & \vec{r}=(\hat{i}+2 \hat{j}) A \cos \omega t \\ & x=A \cos \omega t \\ & y=2 A \cos \omega t \\ & y=2 x \end{aligned}$$

The path is straight line.

The motion is SHM and periodic as

$$\begin{aligned} & \frac{d r}{d t}=-(\hat{i}+2 \hat{j}) \omega A \sin \omega t \\ & \frac{d^2 r}{d t^2}=-(\hat{i}+2 \hat{j}) \omega^2 A \cos \omega t \\ & \vec{a}=-\omega^2 \vec{r} \end{aligned}$$