question_answer

If 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, then the length of the median through A is

A) \[\sqrt{14}\]                

B) \[\sqrt{18}\]    

C) \[\sqrt{29}\]                

D) 4

Correct Answer: B

Solution :

\[\overrightarrow{AD}=\frac{(-3+5)\hat{i}+(0-2)\hat{j}+(4+4)\hat{k}}{2}\]

\[=\frac{2\hat{i}-2\hat{j}+8\hat{k}}{2}=\hat{i}-\hat{j}+4\hat{k}\]

\[\therefore \]  length of median

\[=\left| \overrightarrow{AD} \right|=\sqrt{{{\left( 1 \right)}^{2}}+{{\left( -1 \right)}^{2}}+{{\left( 4 \right)}^{2}}}=\sqrt{18}\]