The position vectors of the points A and B are \(\vec{a}\) and \(\vec{b}\) respectively. C divides AB internally and D divides AB externally, both in the ratio 2 : 3. The position vector of the mid-point of CD is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( \displaystyle \frac{9\vec{a}-4\vec{b}}{5} \)
By the section formula for internal division in the ratio 2 : 3:
\( \displaystyle \vec{c} = \frac{2\vec{b}+3\vec{a}}{2+3} = \frac{3\vec{a}+2\vec{b}}{5} \)
For external division in the ratio 2 : 3:
\( \displaystyle \vec{d} = \frac{2\vec{b}-3\vec{a}}{2-3} = 3\vec{a}-2\vec{b} \)
The mid-point of CD has position vector
\( \displaystyle \frac{\vec{c}+\vec{d}}{2} = \frac{1}{2}\left(\frac{3\vec{a}+2\vec{b}}{5}+\frac{15\vec{a}-10\vec{b}}{5}\right) \)
\( \displaystyle = \frac{18\vec{a}-8\vec{b}}{10} = \frac{9\vec{a}-4\vec{b}}{5} \)
Why the other options are incorrect
Option (1) \(\frac{9\vec{a}+4\vec{b}}{5}\) — Incorrect. The sign of \(\vec{b}\) is wrong: the \(\vec{b}\)-part is \(\frac{1}{2}\left(\frac{2}{5}-2\right)\vec{b} = -\frac{4}{5}\vec{b}\).
Option (2) \(3\vec{a}-2\vec{b}\) — Incorrect. This is the position vector of D, not of the mid-point of CD.
Option (3) \(\frac{3\vec{a}+2\vec{b}}{5}\) — Incorrect. This is the position vector of C, not of the mid-point of CD.