Question

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 a triangle $ABC$. The length of the median through $A$ is

Answer 1

Answer: 5

Let the given vectors be $\overrightarrow{AB}=3\hat{i}+5\hat{j}+4\hat{k}$ and $\overrightarrow{AC}=5\hat{i}-5\hat{j}+2\hat{k}$

Let $AM$ be the median through $A$
$\therefore \overrightarrow{AM}=\frac{1}{2}(\overrightarrow{AB}+\overrightarrow{AC})$
$=\frac{1}{2}[(3\hat{i}+5\hat{j}+4\hat{k})+(5\hat{i}-5\hat{j}+2\hat{k})]$
$=\frac{1}{2}(8\hat{i}+6\hat{k})=(4\hat{i}+3\hat{k})$
$\therefore$ Length of the median $AM$
$=\sqrt{4^2+3^2}=\sqrt{25}=5$ units