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

The correct answer is Option 2. Full mathematical solution steps and hints are accessible in the interactive StemCET workspace.

Maths Q16 – JEE-MAIN 2026

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Home Maths Vector AlgebraQuestion #511

Let O be the origin,

\vec{O P} = \vec{a}

and

\vec{O Q} = \vec{b}

. If R is the point on OP such that

\vec{O P} = 5 \vec{O R}

, and M is the point such that

\vec{O Q} = 5 \vec{R M}

, then

\vec{P M}

is equal to :

Clue 1 of 3: Vector Representation

Express all given vector relationships in terms of position vectors relative to the origin O.

🧠 Full Solution Path

Step 1: Determine Position Vector of R✦ Active

Given that O is the origin, $\vec{O P} = \vec{a}$ and $\vec{O Q} = \vec{b}$ . R is a point on OP such that $\vec{O P} = 5 \vec{O R}$ . This implies that the position vector of R, $\vec{r}$ , is related to the position vector of P, $\vec{p}$ , by $\vec{p} = 5 \vec{r}$ . Since $\vec{p} = \vec{a}$ , we have:

\vec{a} = 5 \vec{r} ⟹ \vec{r} = \frac{1}{5} \vec{a}

Step 2: Determine Position Vector of M○ Expand

M is a point such that $\vec{O Q} = 5 \vec{R M}$ . We know $\vec{O Q} = \vec{b}$ and $\vec{R M} = \vec{m} - \vec{r}$ , where $\vec{m}$ is the position vector of M. Substituting these and the expression for $\vec{r}$ from Step 1:

\vec{b} = 5 (\vec{m} - \vec{r})

\vec{b} = 5 (\vec{m} - \frac{1}{5} \vec{a})

\vec{b} = 5 \vec{m} - \vec{a}

5 \vec{m} = \vec{a} + \vec{b}

\vec{m} = \frac{1}{5} (\vec{a} + \vec{b})

Step 3: Calculate Vector PM○ Expand

The vector $\vec{P M}$ is given by the difference of the position vectors of M and P, i.e., $\vec{P M} = \vec{m} - \vec{p}$ . We have $\vec{p} = \vec{a}$ and $\vec{m} = \frac{1}{5} (\vec{a} + \vec{b})$ . Substituting these values:

\vec{P M} = \frac{1}{5} (\vec{a} + \vec{b}) - \vec{a}

\vec{P M} = \frac{1}{5} \vec{a} + \frac{1}{5} \vec{b} - \vec{a}

\vec{P M} = (\frac{1}{5} - 1) \vec{a} + \frac{1}{5} \vec{b}

\vec{P M} = (\frac{1 - 5}{5}) \vec{a} + \frac{1}{5} \vec{b}

\vec{P M} = - \frac{4}{5} \vec{a} + \frac{1}{5} \vec{b}

\vec{P M} = \frac{1}{5} (\vec{b} - 4 \vec{a})

Comparing this result with the given options, it matches option 2.

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Maths Question 16 – JEE-MAIN 2026

Clue 1 of 3: Vector Representation

🧠 Full Solution Path