The position vectors of the points P and Q are \(3\vec{a}-\vec{b}\) and \(\vec{a}+2\vec{b}\) respectively. The position vector of the point R which divides PQ externally in the ratio 3 : 2 is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(-3\vec{a}+8\vec{b}\)
For a point dividing PQ externally in the ratio m : n, the section formula gives
\( \displaystyle \vec{r} = \frac{m\,\overrightarrow{OQ}-n\,\overrightarrow{OP}}{m-n} \)
Here m = 3 and n = 2, so m − n = 1:
\(\vec{r} = 3(\vec{a}+2\vec{b})-2(3\vec{a}-\vec{b})\)
\( = 3\vec{a}+6\vec{b}-6\vec{a}+2\vec{b}\)
\( = -3\vec{a}+8\vec{b}\)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{9\vec{a}+4\vec{b}}{5}\) — Incorrect. This is \(\frac{3\overrightarrow{OQ}+2\overrightarrow{OP}}{5}\), the point dividing PQ internally in the ratio 3 : 2.
Option (2) \(3\vec{a}-8\vec{b}\) — Incorrect. It is the negative of the correct vector, a sign slip.
Option (3) \(7\vec{a}-7\vec{b}\) — Incorrect. This is \(3\overrightarrow{OP}-2\overrightarrow{OQ}\), with the two points interchanged.