Question

The two vectors $\hat{j}+\hat{k}$ and $3\hat{i}-\hat{j}+4\hat{k}$ represent the two sides $AB$ and $AC$, respectively of a $△ABC$. Find the length of the median through $A$

Answer 1

Let $A$ be the origin.

Thus, we get the position vectors of points $B$ and $C$ as $\overrightarrow{b}=\hat{j}+\hat{k}$ and $\overrightarrow{c}=3\hat{i}-\hat{j}+4\hat{k}$

Now, the midpoint of $BC$ would thus be $\frac{\overrightarrow{b}+\overrightarrow{c}}{2}$

i.e. $\frac{\hat{j}+\hat{k}+3\hat{i}-\hat{j}+4\hat{k}}{2}$

$=\frac{3\hat{i}+5\hat{k}}{2}$

The length of the median through $A$ thus becomes $\sqrt{\frac{9+25}{4}}=\frac{\sqrt{34}}{2}$