Let $$O$$ be the origin, $$\overrightarrow{OP} = \vec{a}$$ and $$\overrightarrow{OQ} = \vec{b}$$.If $$R$$ is the point on $$\overrightarrow{OP}$$ such that $$\overrightarrow{OP} = 5\overrightarrow{OR}$$, and $$M$$ is the point such that $$\overrightarrow{OQ} = 5\overrightarrow{RM}$$. Then $$\overrightarrow{PM}$$ is equal to :

$$\vec{PM} = \vec{OM} - \vec{OP}$$

$$\vec{OP} = 5\vec{OR}$$

$$\vec{OR} = \frac{1}{5}\vec{OP} = \frac{1}{5}\vec{a}$$

$$\vec{OQ} = 5\vec{RM}$$

$$\vec{OQ} = 5(\vec{OM} - \vec{OR})$$

$$\vec{b} = 5\left(\vec{OM} - \frac{1}{5}\vec{a}\right)$$

$$\vec{b} = 5\vec{OM} - \vec{a}$$

$$5\vec{OM} = \vec{b} + \vec{a} \implies \vec{OM} = \frac{1}{5}(\vec{b} + \vec{a})$$

$$\vec{PM} = \frac{1}{5}(\vec{b} + \vec{a}) - \vec{a}$$

$$\vec{PM} = \frac{1}{5}(\vec{b} - 4\vec{a})$$