Question

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

• A. \( \frac{4\vec{b} - \vec{a}}{5} \)
• B. \( \frac{\vec{b} - 4\vec{a}}{5} \)
• C. \( \frac{5\vec{b} - \vec{a}}{4} \)
• D. \( \frac{\vec{b} - 5\vec{a}}{4} \)

Hint:

In vector geometry, always express unknown vectors in terms of position vectors relative to the origin (e.g., \( \vec{AB} = \vec{b} - \vec{a} \)) to simplify the algebra.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We use vector addition and the triangle law of vectors. The position vectors of points \( P, Q, R, \) and \( M \) are defined relative to the origin \( O \).

Step 2: Key Formula or Approach:
1. \( \vec{PM} = \vec{OM} - \vec{OP} \). 2. Find \( \vec{OM} \) using the given relation for \( \vec{RM} \).

Step 3: Detailed Explanation:
1. Given: \( \vec{OP} = \vec{a} \), \( \vec{OQ} = \vec{b} \). 2. \( \vec{OR} = \frac{1}{5} \vec{OP} = \frac{\vec{a}}{5} \). 3. Using triangle law for \( \Delta ORM \): \( \vec{OM} = \vec{OR} + \vec{RM} \). 4. Given \( \vec{RM} = \frac{1}{5} \vec{OQ} = \frac{\vec{b}}{5} \). 5. Substitute: \( \vec{OM} = \frac{\vec{a}}{5} + \frac{\vec{b}}{5} \). 6. Now, find \( \vec{PM} \): \[ \vec{PM} = \vec{OM} - \vec{OP} = \left( \frac{\vec{a}}{5} + \frac{\vec{b}}{5} \right) - \vec{a} \] \[ \vec{PM} = \frac{\vec{a} + \vec{b} - 5\vec{a}}{5} = \frac{\vec{b} - 4\vec{a}}{5} \]

Step 4: Final Answer:
The vector \( \vec{PM} \) is \( \frac{\vec{b} - 4\vec{a}}{5} \).