A and B are points with position vectors \(\vec{a}\) and \(\vec{b}\) respectively. The line segment AB is extended beyond B to a point C such that \(AC = 3\,AB\). The position vector of C is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(3\vec{b}-2\vec{a}\)
Since C lies on AB extended beyond B, the vectors \(\overrightarrow{AC}\) and \(\overrightarrow{AB}\) have the same direction, and \(AC = 3\,AB\). So
\(\overrightarrow{AC} = 3\,\overrightarrow{AB}\)
Writing each vector as (end) minus (start) in position vectors, with \(\vec{c}\) for C:
\(\vec{c}-\vec{a} = 3(\vec{b}-\vec{a})\)
\(\vec{c} = 3\vec{b}-2\vec{a}\)
This is the same as saying C divides AB externally in the ratio 3 : 2.
Why the other options are incorrect
Option (1) \(3\vec{b}-\vec{a}\) — Incorrect. It gives \(\overrightarrow{AC} = 3\vec{b}-2\vec{a}\), but \(3\,\overrightarrow{AB} = 3\vec{b}-3\vec{a}\).
Option (2) \(2\vec{a}-3\vec{b}\) — Incorrect. This is the negative of the correct vector, a point on the other side of the origin.
Option (4) \(\frac{\vec{a}+3\vec{b}}{4}\) — Incorrect. This point divides AB internally in the ratio 3 : 1, so it lies between A and B.