Point S divides PQ in the ratio 5:2. Express OS in terms of p and q.

a) Express in terms of $\mathbf{p}$ and $\mathbf{q}$.

i) $\vec{OS}$
Point S divides PQ in the ratio 5:2. This means $\vec{PS} = \frac{5}{7}\vec{PQ}$.
First, find $\vec{PQ}$:

$$\vec{PQ} = \vec{OQ} - \vec{OP} = \mathbf{q} - \mathbf{p}$$

Now, find $\vec{OS}$:

$$\vec{OS} = \vec{OP} + \vec{PS}$$ $$$$