VectorMedium

Question

Let O be the origin, $\overrightarrow{OP} = \overrightarrow{a}$ and $\overrightarrow{OQ} = \overrightarrow{b}$. 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 :

Options

A.$\frac{1}{5}(\overrightarrow{a} - 4\overrightarrow{b})$
B.$\frac{1}{5}(\overrightarrow{b} - 4\overrightarrow{a})$
C.$\frac{1}{5}( - \overrightarrow{a} + 4\overrightarrow{b})$
D.$\frac{1}{5}( - \overrightarrow{b} + 4\overrightarrow{a})$

Solution

Sol. [Key Concept: With $OR = \frac{a}{5}$ and $RM = \frac{b}{5}$, $OM = OR + RM$; then $PM = OM - OP$.]

$${\overrightarrow{OR} = \frac{\overrightarrow{OP}}{5} = \frac{\overrightarrow{a}}{5} }{\overrightarrow{RM} = \frac{\overrightarrow{OQ}}{5} = \frac{\overrightarrow{b}}{5} }{\overrightarrow{OM} - \overrightarrow{OR} = \frac{\overrightarrow{b}}{5} }{\overrightarrow{OM} = \frac{\overline{a} + \overline{b}}{5} }{\overrightarrow{PM} = \overrightarrow{OM} - \overrightarrow{OP} }{\frac{\overrightarrow{a} + \overrightarrow{b}}{5} - \overrightarrow{a} }{= \frac{\overrightarrow{b} - 4\overrightarrow{a}}{5} }$$

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