The point R with position vector \(3\hat{\imath}+\hat{k}\) divides the line segment PQ internally in the ratio 2 : 3. If the position vector of P is \(\hat{\imath}+2\hat{\jmath}-3\hat{k}\), then the position vector of Q is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(6\hat{\imath}-3\hat{\jmath}+7\hat{k}\)
Let \(\vec{p}, \vec{q}, \vec{r}\) be the position vectors of P, Q, R. Since R divides PQ internally in the ratio 2 : 3, the section formula gives
\( \displaystyle \vec{r} = \frac{2\vec{q}+3\vec{p}}{5} \)
\(2\vec{q} = 5\vec{r}-3\vec{p}\)
\(2\vec{q} = (15\hat{\imath}+5\hat{k})-(3\hat{\imath}+6\hat{\jmath}-9\hat{k})\)
\(2\vec{q} = 12\hat{\imath}-6\hat{\jmath}+14\hat{k}\)
\(\vec{q} = 6\hat{\imath}-3\hat{\jmath}+7\hat{k}\)
Why the other options are incorrect
Option (1) \(5\hat{\imath}-2\hat{\jmath}+5\hat{k}\) — Incorrect. This is \(2\vec{r}-\vec{p}\), which treats R as the mid-point of PQ.
Option (2) \(\frac{13}{3}\hat{\imath}-\frac{4}{3}\hat{\jmath}+\frac{11}{3}\hat{k}\) — Incorrect. This comes from the reversed ratio, \(\vec{r} = \frac{3\vec{q}+2\vec{p}}{5}\).
Option (3) \(4\hat{\imath}+2\hat{\jmath}-2\hat{k}\) — Incorrect. This is \(\vec{p}+\vec{r}\), which is not given by the section formula.