मान लीजिए \(O\) मूल-बिंदु है, \(\vec{OP} = \vec{a}\) और \(\vec{OQ} = \vec{b}\)। यदि \(R\) एक ऐसा बिंदु है जो \(\vec{OP}\) पर स्थित है और \(\vec{OP} = 5\vec{OR}\) है, तथा \(M\) एक ऐसा बिंदु है कि \(\vec{OQ} = 5\vec{RM}\) है, तो \(\vec{PM}\) बराबर है :

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

Verified Solution

Answer & Solution

Correct Answer

(B) \(\dfrac{1}{5}(\vec{b} - 4\vec{a})\)

Step-by-step Solution

Detailed explanation

दिया है \(\vec{OP} = \vec{a}\) और \(\vec{OQ} = \vec{b}\)। चूंकि \(\vec{OP} = 5\vec{OR}\), अतः हमें प्राप्त होता है \(\vec{OR} = \dfrac{\vec{a}}{5}\)। साथ ही, \(\vec{OQ} = 5\vec{RM}\), जिससे प्राप्त होता है \(\vec{RM} = \dfrac{\vec{b}}{5}\)। \(M\) का स्थिति सदिश इस प्रकार दिया…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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