A) \[\frac{2}{3}(-6\hat{i}-8\hat{j}-6\hat{k})\]
B) \[\frac{2}{3}(6\hat{i}+8\hat{j}+6\hat{k})\]
C) \[\frac{1}{3}(6\hat{i}+8\hat{j}\,+18\hat{k})\]
D) \[\frac{1}{3}(5\hat{j}+12\hat{k})\]
Correct Answer: C
Solution :
[c] Suppose the bisector of angle A meets BC at D. Then AD divides BC in the ratio AB: AC. So, P.V. of D is given by \[\frac{\overrightarrow{\left| AB \right|}(2\hat{i}+5\hat{j}+7\hat{k})+\overrightarrow{\left| AC \right|}(2\hat{i}+3\hat{j}+4\hat{k})}{\overrightarrow{\left| AB \right|}+\overrightarrow{\left| AC \right|}}\] But \[\overrightarrow{AB}=-2\hat{i}-4\hat{j}-4\hat{k}\] And \[\overrightarrow{AC}=-2\hat{i}-2\hat{j}-\hat{k}\] \[\Rightarrow \overrightarrow{\left| AB \right|}=6\] and \[\overrightarrow{\left| AC \right|}=3\] Therefore, P.V. of D is given by \[\frac{6(2\hat{i}+5\hat{j}+7\hat{k})+3(2\hat{i}+3\hat{j}+4\hat{k})}{6+3}\] \[=\frac{1}{3}(6\hat{i}+13\hat{j}+18\hat{k})\]
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