In a simple harmonic motion, which of the following quantities is...

In simple harmonic motion, the restoring force $\vec{F}$ and acceleration $\vec{a}$ are related by Newton's second law as $\vec{F} = m\vec{a}$.

Taking the dot product of $\vec{F}$ and $\vec{a}$, we get:

$\vec{F} \cdot \vec{a} = (m\vec{a}) \cdot \vec{a} = m|\vec{a}|^2 = ma^2$

Since mass $m$ is a positive quantity and the square of the magnitude of acceleration $a^2$ is always non-negative, their product is always positive (or zero at the mean position).

For the other options:

$\vec{a} \cdot \vec{r} = (-\omega^2 \vec{r}) \cdot \vec{r} = -\omega^2 r^2 \le 0$

$\vec{F} \cdot \vec{r} = (-k \vec{r}) \cdot \vec{r} = -k r^2 \le 0$

$\vec{v} \cdot \vec{r}$ can be positive or negative depending on whether the particle is moving away from or towards the mean position.