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

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