Point S divides the line segment PQ in the ratio 5:2. Express in terms of p and q: a) OS i) vector OS

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

i) $\vec{OS}$
Point S divides the line segment PQ in the ratio 5:2.
Using the section formula for position vectors:

$$\vec{OS} = \frac{2\vec{OP} + 5\vec{OQ}}{5+2}$$ $$\vec{OS} = \frac{2\mathbf{p} + 5\mathbf{q}}{7}$$ $$\vec{OS} = \frac{2}{7}\mathbf{p} + \frac{5}{7}\mathbf{q}$$

The vector $\vec{OS}$ is $\boxed