question_answer

The     vectors     \[\overrightarrow{AB}=3\hat{i}+4\hat{k}\]    and \[\overrightarrow{AC}=5\hat{i}-2\hat{j}+4\hat{k}\] are the sides of a triangle ABC. The length of the median through A is

A)  \[\sqrt{18}\]          

B)  \[\sqrt{72}\]

C)  \[\sqrt{33}\]          

D)  \[\sqrt{288}\]

Correct Answer: C

Solution :

Given vectors are \[\overrightarrow{AB}=3\hat{i}+4\hat{k}\] and \[\overrightarrow{AC}=5\hat{i}-2\hat{j}+4\hat{k}\] Equation of median, \[\overrightarrow{AD}=\frac{\overrightarrow{AB}+\overrightarrow{AC}}{2}=\frac{(3\hat{i}+4\hat{k})+(5-2\hat{j}+4\hat{k})}{2}\] \[=4\hat{i}-\hat{j}+4\hat{k}\] \[\therefore \]  Length of median, \[|\overrightarrow{AD}|=\sqrt{{{(4)}^{2}}+{{(-1)}^{2}}+{{(4)}^{2}}}\] \[\sqrt{16+1+16}=\sqrt{33}\]