Finding height of a triangle with vectors

Question

I'm given a triangle with coordinates $A(3,6)$, $B(-1,3)$ and $C(2,-1)$. Find the length of height of that triangle. Well I know how to find it with constructing some lines normal to sides of that triangle, finding intersection etc. and calculating lengths, but is there a trick to do it easier with vectors? Like using vector or scalar product for example?

Answer 1

Find the vector $AB$ and the vector $AC$. Recall that $||AB \times AC||$ represents the area of the parallelogram with the sides equal in lengths and direction to the vectors $AB$ and $AC$ (here ||<>|| denotes magnitude and $\times$ denotes the cross product). Take $AB$ to be the base then using the fact $(b)(h)=\text{area}$ we have:

$$h=\frac{\text{area}}{b}=\frac{||AB \times AC||}{||AB||}$$

The height in the parallelogram is the same height as in the triangle with the same sides, if you take the base to be the same.

Note:

The cross product may not make that much sense in $2-D$ geometrically, so work in $3-D$. Just take the components of the vectors $AB$ and $AC$ along the $z$ axis to be $0$ and then work like usual.