question_answer

If the vectors\[\overrightarrow{AB}=3\hat{i}+5\hat{j}+4\hat{k}\]and\[\overrightarrow{Ac}=5\hat{i}-5\hat{j}+2\hat{k}\]are the sides of the\[\Delta ABC,\]then length of median passing from A is

A)  \[\sqrt{3}\]units          

B) \[2\sqrt{5}\]units

C)  5 units           

D)  10 units

Correct Answer: C

Solution :

 Given, vector \[\overrightarrow{AB}=3\hat{i}+5\hat{j}+4\hat{k}\] and vector \[\overrightarrow{AC}=5\hat{i}-5\hat{j}+2\hat{k}\] Let AM is the median passing through the point A. \[\therefore \] \[\overrightarrow{AM}=\frac{1}{2}(\overrightarrow{AB}+\overrightarrow{AC})\] \[=\frac{1}{2}[(3\hat{i}+5\hat{j}+4\hat{k})+(5\hat{i}-5\hat{j}+2\hat{k})]\] \[=\frac{1}{2}[8\hat{i}+6\hat{k}]\] \[=4\hat{i}+3\hat{k}\] Length of median \[AM=\sqrt{{{4}^{2}}+{{3}^{2}}}\] \[=\sqrt{16+9}=\sqrt{25}=5\]units