question_answer

If the middle points of sides BC, CA & AB of triangle ABC are respectively D, E, F then position vector of centre of triangle DEF, when position vector of A, B, C are respectively \[\hat{i}+\hat{j},\hat{j}+\hat{k},\hat{k}+\hat{i}\]is

A) \[\frac{1}{3}(\hat{i}+\hat{j}+\hat{k})\]

B) \[(\hat{i}+\hat{j}+\hat{k})\]

C) \[2(\hat{i}+\hat{j}+\hat{k})\]

D) \[\frac{2}{3}(\hat{i}+\hat{j}+\hat{k})\]

Correct Answer: D

Solution :

[d] The position vector of points D, E, F are respectively \[\frac{\hat{i}+\hat{j}}{2}+\hat{k},\hat{i}+\frac{\hat{k}+\hat{j}}{2}\] and \[\frac{\hat{i}+\hat{k}}{2}+\hat{j}\] So, position vector of centre of \[\Delta DEF\] \[=\frac{1}{3}\left[ \frac{\hat{i}+\hat{j}}{2}+\hat{k}+\hat{i}\frac{\hat{k}+\hat{j}}{2}+\frac{\hat{i}+\hat{k}}{2}+\hat{j} \right]\] \[=\frac{2}{3}\left[ \hat{i}+\hat{j}+\hat{k} \right]\]