The position vector of a point which divides the line joining the points with position vectors \((\vec{a} - 2\vec{b})\) and \((2\vec{a} + \vec{b})\) externally in the ratio 2 : 1, is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(3\vec{a} + 4\vec{b}\)
By the section formula for external division, the point dividing the join of P(\(\vec{p}\)) and Q(\(\vec{q}\)) externally in the ratio m : n has position vector:
\( \displaystyle \vec{r} = \frac{m\vec{q} - n\vec{p}}{m - n} \)
Here \(\vec{p} = \vec{a} - 2\vec{b}\), \(\vec{q} = 2\vec{a} + \vec{b}\), m = 2 and n = 1:
\( \displaystyle \vec{r} = \frac{2(2\vec{a} + \vec{b}) - (\vec{a} - 2\vec{b})}{2 - 1} \)
\(= 4\vec{a} + 2\vec{b} - \vec{a} + 2\vec{b}\)
\(= 3\vec{a} + 4\vec{b}\)
Why the other options are incorrect
Option (2) \(-3\vec{a} + 4\vec{b}\) — Incorrect. The coefficient of \(\vec{a}\) has the wrong sign.
Option (3) \(3\vec{a} - 4\vec{b}\) — Incorrect. The coefficient of \(\vec{b}\) has the wrong sign.
Option (4) \(\displaystyle\frac{5}{3}\vec{a}\) — Incorrect. This is the point of internal division in the ratio 2 : 1: \( \frac{2(2\vec{a} + \vec{b}) + (\vec{a} - 2\vec{b})}{3} = \frac{5}{3}\vec{a} \).
NCERT (2026-27 edition):
Section Formula
