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 define the position vectors of points P, Q, R, and M relative to the origin O.
Step 2: Detailed Explanation:
Given \( \vec{OP} = \vec{a} \) and \( \vec{OQ} = \vec{b} \).
From \( 5\vec{OR} = \vec{OP} \), we have \( \vec{OR} = \frac{\vec{a}}{5} \).
From \( 5\vec{RM} = \vec{OQ} \), we have \( 5(\vec{OM} - \vec{OR}) = \vec{b} \).
Substituting \( \vec{OR} \):
\[ 5\left(\vec{OM} - \frac{\vec{a}}{5}\right) = \vec{b} \implies 5\vec{OM} - \vec{a} = \vec{b} \implies 5\vec{OM} = \vec{a} + \vec{b} \implies \vec{OM} = \frac{\vec{a} + \vec{b}}{5} \]
We need to find \( \vec{PM} \):
\[ \vec{PM} = \vec{OM} - \vec{OP} = \frac{\vec{a} + \vec{b}}{5} - \vec{a} \]
\[ \vec{PM} = \frac{\vec{a} + \vec{b} - 5\vec{a}}{5} = \frac{\vec{b} - 4\vec{a}}{5} \]
Step 4: Final Answer:
The vector \( \vec{PM} \) is \( \frac{\vec{b} - 4\vec{a}}{5} \).