Vector Position Expression for PM

To find the vector expression for $\vec{PM}$, we first determine the position vectors of the relevant points relative to the origin $O$.

Given:

1. Position vector of $P$: $\vec{OP} = \vec{a}$

2. Position vector of $Q$: $\vec{OQ} = \vec{b}$

3. Point $R$ lies on $\vec{OP}$ such that $\vec{OP} = 5\vec{OR}$: $\vec{OR} = \frac{1}{5}\vec{OP} = \frac{1}{5}\vec{a}$

4. Point $M$ satisfies $\vec{OQ} = 5\vec{RM}$: $\vec{RM} = \frac{1}{5}\vec{OQ} = \frac{1}{5}\vec{b}$

Using triangle law of vector addition, the position vector of point $M$ relative to the origin $O$ is: $\vec{OM} = \vec{OR} + \vec{RM}$

Substitute $\vec{OR}$ and $\vec{RM}$ into the equation: $\vec{OM} = \frac{1}{5}\vec{a} + \frac{1}{5}\vec{b} = \frac{1}{5}(\vec{a} + \vec{b})$

Now, the vector $\vec{PM}$ can be expressed in terms of position vectors $\vec{OM}$ and $\vec{OP}$: $\vec{PM} = \vec{OM} - \vec{OP}$

Substituting the expressions for $\vec{OM}$ and $\vec{OP}$: $\vec{PM} = \frac{1}{5}(\vec{a} + \vec{b}) - \vec{a}$ $\vec{PM} = \frac{1}{5}\vec{a} + \frac{1}{5}\vec{b} - \vec{a}$ $\vec{PM} = \frac{1}{5}\vec{b} - \frac{4}{5}\vec{a}$ $\vec{PM} = \frac{1}{5}(\vec{b} - 4\vec{a})$

Thus, the vector $\vec{PM}$ is equal to $\frac{1}{5}(\vec{b} - 4\vec{a})$.

Correct Option: B