Q41 / 75 · JEE Main 2026 (05 April Shift 2) · Mathematics

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

  • A
    $\dfrac{1}{5}(\vec{a} - 4\vec{b})$
  • Correct answer
    B
    $\dfrac{1}{5}(\vec{b} - 4\vec{a})$
  • C
    $\dfrac{1}{5}(-\vec{a} + 4\vec{b})$
  • D
    $\dfrac{1}{5}(-\vec{b} + 4\vec{a})$

Solution

Given $\vec{OP} = \vec{a}$ and $\vec{OQ} = \vec{b}$.

Since $\vec{OP} = 5\vec{OR}$, we have $\vec{OR} = \dfrac{\vec{a}}{5}$.

Also, $\vec{OQ} = 5\vec{RM}$, which gives $\vec{RM} = \dfrac{\vec{b}}{5}$.

The position vector of $M$ is given by $\vec{OM} = \vec{OR} + \vec{RM} = \dfrac{\vec{a}}{5} + \dfrac{\vec{b}}{5}$.

Now, we need to find $\vec{PM}$:

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

$\vec{PM} = \left(\dfrac{\vec{a}}{5} + \dfrac{\vec{b}}{5}\right) - \vec{a}$

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

Answer: $\dfrac{1}{5}(\vec{b} - 4\vec{a})$

Hence, the correct answer is option: 'B'.

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