The position vectors of the points A and B are \(\hat{\imath}+2\hat{\jmath}-\hat{k}\) and \(4\hat{\imath}-\hat{\jmath}+5\hat{k}\) respectively. The position vector of the point which divides AB internally in the ratio 1 : 2 is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(2\hat{\imath}+\hat{\jmath}+\hat{k}\)
Let \(\vec{a}\) and \(\vec{b}\) be the position vectors of A and B. By the section formula, the point dividing AB internally in the ratio m : n has position vector
\( \displaystyle \vec{r} = \frac{m\vec{b}+n\vec{a}}{m+n} \)
Here m = 1 and n = 2:
\( \displaystyle \vec{r} = \frac{\vec{b}+2\vec{a}}{3} \)
\(\vec{b}+2\vec{a} = (4+2)\hat{\imath}+(-1+4)\hat{\jmath}+(5-2)\hat{k} = 6\hat{\imath}+3\hat{\jmath}+3\hat{k}\)
\( \displaystyle \vec{r} = \frac{6\hat{\imath}+3\hat{\jmath}+3\hat{k}}{3} = 2\hat{\imath}+\hat{\jmath}+\hat{k} \)
Why the other options are incorrect
Option (1) \(3\hat{\imath}+3\hat{k}\) — Incorrect. This is \(\frac{2\vec{b}+\vec{a}}{3}\), the point dividing AB in the ratio 2 : 1.
Option (3) \(-2\hat{\imath}+5\hat{\jmath}-7\hat{k}\) — Incorrect. This is \(\frac{\vec{b}-2\vec{a}}{1-2}\), the point dividing AB externally in the ratio 1 : 2.
Option (4) \(\displaystyle\frac{5}{2}\hat{\imath}+\frac{1}{2}\hat{\jmath}+2\hat{k}\) — Incorrect. This is the mid-point of AB, which divides it in the ratio 1 : 1.