Question

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

• A. $\sqrt{13}$ units
• B. $2\sqrt{5}$ units
• C. $5$ units
• D. $10$ units

Answer 1

The correct option is C $5$ units

Given

$\overrightarrow{AB}=3\hat{i}+5\hat{j}+4\hat{k}$

$\overrightarrow{OB}=3\hat{i}+5\hat{j}+4\hat{k}+\overrightarrow{OA}$

$\overrightarrow{AC}=5\hat{i}-5\hat{j}+2\hat{k}$

$\overrightarrow{OC}=5\hat{i}-5\hat{j}+2\hat{k}+\overrightarrow{OA}$

D is the mid point of B,C

So

$\overrightarrow{OD}=\frac{\overrightarrow{OB}+\overrightarrow{OC}}{2}$

$\overrightarrow{OD}=\frac{3\hat{i}+5\hat{j}+4\hat{k}+\overrightarrow{OA}+5\hat{i}-5\hat{j}+2\hat{k}+\overrightarrow{OA}}{2}$

$\overrightarrow{OD}=\frac{8\hat{i}+6\hat{k}+2\overrightarrow{OA}}{2}$

$\overrightarrow{OD}=4\hat{i}+3\hat{k}+\overrightarrow{OA}$

$\overrightarrow{OD}-\overrightarrow{OA}=4\hat{i}+3\hat{k}$

$\overrightarrow{AD}=4\hat{i}+3\hat{k}$

length of AD

$\left|\overrightarrow{AD}\right|=\sqrt{4^2+3^2}$

$\left|\overrightarrow{AD}\right|=\sqrt{16+9}$

$\left|\overrightarrow{AD}\right|=\sqrt{25}$

$\left|\overrightarrow{AD}\right|=5$

Hence option C is correct