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:

Here's how to express $\vec{QS}$ in terms of $\mathbf{p}$ and $\mathbf{r}$:

Step 1: Express $\vec{OQ}$ in terms of $\mathbf{p}$ and $\mathbf{r}$. 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 $\vec{OR} = \mathbf{r}$: $\vec{OQ} = \frac{2p + r}{3}$

Step 2: Express $\vec{OS}$ in terms of $\mathbf{p}$ and $\mathbf{r}$. 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 $\vec{OR} = \mathbf{r}$: $\vec{OS} - p = 3(r - p)$ $\vec{OS} - p = 3r - 3p$ $\vec{OS} = 3r - 3p + p$ $\vec{OS} = 3r - 2p$

Step 3: Calculate $\vec{QS}$. 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: $\vec{QS} = (3r - 2p) - \left(\frac{2p + r}{3}\right)$ $\vec{QS} = 3r - 2p - \frac{2}{3}p - \frac{1}{3}r$ Combine the terms with $\mathbf{p}$ and $\mathbf{r}$: $\vec{QS} = \left(3 - \frac{1}{3}\right)r + \left(-2 - \frac{2}{3}\right)p$ $\vec{QS} = \left(\frac{9}{3} - \frac{1}{3}\right)r + \left(-\frac{6}{3} - \frac{2}{3}\right)p$ $\vec{QS} = \frac{8}{3}r - \frac{8}{3}p$ \vec{QS} = \frac{8{3}(r - p)}

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