During a simple harmonic motion, which of the following quantities...

In simple harmonic motion, the restoring force $\vec{F}$ and acceleration $\vec{a}$ are always directed towards the mean position, while the displacement $\vec{r}$ is directed away from the mean position.

Mathematically, $\vec{F} = -k\vec{r}$ and $\vec{a} = -\omega^2\vec{r}$, where $k$ and $\omega^2$ are positive constants.

Evaluating the given dot products:

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

$\vec{v} \cdot \vec{r}$ can be positive (when the particle is moving away from the mean position) or negative (when moving towards the mean position).

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

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

Thus, $\vec{F} \cdot \vec{r}$ and $\vec{a} \cdot \vec{r}$ will always be negative for any non-zero displacement.