If $\vec{AB} = \vec b$ and $\vec{AC} =\vec c$, then the length of the perpendicular from A to the line BC is

$\frac{|\vec b×\vec c|}{|\vec b-\vec c|}$

We have,

Area of ΔABC = $\frac{1}{2}|\vec{AB}×\vec{AC}|=|\vec b×\vec c|$

Also,

Area of ΔABC = $\frac{1}{2}\{\vec{BC}$ × Length of the ⊥ from A and BC}

∴ Required length = $\frac{|\vec b×\vec c|}{|\vec{BC}|}=\frac{|\vec b×\vec c|}{|\vec b-\vec c|}$