To solve this problem, we need to analyze the motion of the particle given by the position vector $\vec{r} = x \cos(\omega t) \hat{i} + y \sin(\omega t) \hat{j}. \\$First, let's find the velocity vector $\vec{v}. \\$The velocity $\vec{v}$ is the time derivative of the position vector $\vec{r}. \\ \vec{v} = \frac{d\vec{r}}{dt} = \frac{d}{dt}(x \cos(\omega t) \hat{i} + y \sin(\omega t) \hat{j}) \\$Using the chain rule, we get:$\\ \vec{v} = -x \omega \sin(\omega t) \hat{i} + y \omega \cos(\omega t) \hat{j} \\$Next, let's find the acceleration vector $\vec{a}. \\$The acceleration $\vec{a}$ is the time derivative of the velocity vector $\vec{v}. \\ \vec{a} = \frac{d\vec{v}}{dt} = \frac{d}{dt}(-x \omega \sin(\omega t) \hat{i} + y \omega \cos(\omega t) \hat{j}) \\$Again, using the chain rule, we get:$\\ \vec{a} = -x \omega^2 \cos(\omega t) \hat{i} - y \omega^2 \sin(\omega t) \hat{j} \\$Now, let's analyze the direction of $\vec{v}$ and $\vec{a}. \\$1. Velocity $\vec{v}$ is given by:$\\ \vec{v} = -x \omega \sin(\omega t) \hat{i} + y \omega \cos(\omega t) \hat{j} \\$The dot product of $\vec{v}$ and $\vec{r}$ is:$\\ \vec{v} \cdot \vec{r} = (-x \omega \sin(\omega t) \hat{i} + y \omega \cos(\omega t) \hat{j}) \cdot (x \cos(\omega t) \hat{i} + y \sin(\omega t) \hat{j}) \\ = -x^2 \omega \sin(\omega t) \cos(\omega t) + y^2 \omega \cos(\omega t) \sin(\omega t) \\ = \omega \sin(\omega t) \cos(\omega t) (-x^2 + y^2) \\$Since $\sin(\omega t) \cos(\omega t) \neq 0$ and $x^2 \neq y^2,$ the dot product is zero, indicating that $\vec{v}$ is perpendicular to $\vec{r}. \\$2. Acceleration $\vec{a}$ is given by:$\\ \vec{a} = -x \omega^2 \cos(\omega t) \hat{i} - y \omega^2 \sin(\omega t) \hat{j} \\$This can be rewritten as:$\\ \vec{a} = -\omega^2 (x \cos(\omega t) \hat{i} + y \sin(\omega t) \hat{j}) \\ = -\omega^2 \vec{r} \\$This shows that the acceleration $\vec{a}$ is directed towards the origin, as it is opposite to $\vec{r}. \\$Therefore, the correct option is:$\\$Option 1: Velocity is perpendicular to $\vec{r}$ and acceleration is directed towards the origin.$\\$However, the provided correct option is 2, which is incorrect based on the analysis.