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  4. If vectors AB=3 hati-3 hatk and AC= hati-2 hatj+ hatk are the sides of a triangle ABC, then the length of the median AM, is

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Q. If vectors $\overrightarrow{AB}=3\,\hat{i}-3\,\hat{k}$ and $\overrightarrow{AC}=\hat{i}-2\,\hat{j}+\hat{k}$ are the sides of a triangle ABC, then the length of the median AM, is

Vector Algebra

Solution:

If $AM$ is the.medium of $\Delta ABC$ then
$\overrightarrow{AB }+\overrightarrow{AC}=2\,\overrightarrow{AM}$
image
$\Rightarrow 3\,\hat{i}-3\,\hat{k}+\hat{i}-2\,\hat{j}+\hat{k}=2\,\overrightarrow{AM}$
or $2\,\overrightarrow{AM}=4\,\hat{i}-2\,\hat{j}-2\,\hat{k}$
$\therefore \overrightarrow{AM}=2\,\hat{i}-\hat{j}-\hat{k}$
$\therefore \left|\overrightarrow{AM}\right|$ i.e., length $AM=\sqrt{4+1+1}$
$=\sqrt{6}$