Given that PQ:QR = 1:2, point Q divides the line segment PR in the ratio 1:2. Using the section formula for position vectors:

Given that $PQ:QR = 1:2$, point Q divides the line segment PR in the ratio $1:2$. Using the section formula for position vectors:

$$\vec{OQ} = \frac{2\vec{OP} + 1\vec{OR}}{1+2}$$

Substitute $\vec{OP} = \mathbf{p}$ and $\vec{OR} = \mathbf{r}$:

$$\vec{OQ} = \frac{2\mathbf{p} + \mathbf{r}}{3}$$

Given that $PS = 3PR$, this means the vector $\vec{PS}$ is 3 times the vector $\vec{PR}$.

$$\vec{PS} = 3\vec{PR}$$

We know that $\vec{PS} = \vec{OS} - \vec{OP}$ and $\vec{PR} = \vec{OR} - \vec{OP}$.

$$\vec{OS} - \vec{OP} = 3(\vec{OR} - \vec{OP})$$

Substitute $\vec{OP} = \mathbf{p}$ and $\vec{OR} = \mathbf{r}$:

$$\vec{OS} - \mathbf{p} = 3(\mathbf{r} - \mathbf{p})$$ $$\vec{OS} - \mathbf{p} = 3\mathbf{r} - 3\mathbf{p}$$ $$\vec{OS} = 3\mathbf{r} - 3\mathbf{p} + \mathbf{p}$$ $$\vec{OS} = 3\mathbf{r} - 2\mathbf{p}$$

To find $\vec{QS}$, we use the relationship $\vec{QS} = \vec{OS} - \vec{OQ}$.
Substitute the expressions for $\vec{OS}$ and $\vec{OQ}$ from Step 1 and Step 2:

$$\vec{QS} = (3\mathbf{r} - 2\mathbf{p}) - \left(\frac{2\mathbf{p} + \mathbf{r}}{3}\right)$$ $$\vec{QS} = 3\mathbf{r} - 2\mathbf{p} - \frac{2}{3}\mathbf{p} - \frac{1}{3}\mathbf{r}$$

Combine the terms with $\mathbf{p}$ and $\mathbf{r}$:

$$\vec{QS} = \left(3 - \frac{1}{3}\right)\mathbf{r} + \left(-2 - \frac{2}{3}\right)\mathbf{p}$$ $$\vec{QS} = \left(\frac{9}{3} - \frac{1}{3}\right)\mathbf{r} + \left(-\frac{6}{3} - \frac{2}{3}\right)\mathbf{p}$$ $$\vec{QS} = \frac{8}{3}\mathbf{r} - \frac{8}{3}\mathbf{p}$$ $$\vec{QS} = \boxed{\frac{8}{3}(\mathbf{r} - \mathbf{p})}$$