Using vectors, find the height from vertex $A$ of the triangle with vertices $A(-2,-1,-1)$, $B(0,3,2)$, $C(3,3-2)$.

Question

I'm given coordinates, $A(-2,-1,-1)$, $B(0,3,2)$, $C(3,3-2)$. I need to find the height of the triangle from vertex $A$.

Answer 1

If all you need is the length of the altitude and not its foot, recall that the area of a triangle is equal to $\frac12bh$. Compute the area of the triangle (see this question for more ways to do this than you need), divide by the distance between $B$ and $C$ and double the result.

Answer 2

hint

The line $BC$ has parametric equations $$x=x_B+(x_C-x_B)t=0+3t$$ $$y=y_B+(y_C-y_B)t=3$$ $$z=z_B+(z_C-z_B)t=2-4t$$

the point $ H $ of $ BC $ such that $AH$ and $BC$ are perpendicular is given by

$$\vec{AH} . \vec{BC}=0$$ or $$3(3t+2)+0.(3+1)-4(2-4t+1)=0$$ or $$25t-6=0$$

You get $H$ and then the distance

$AH$ which is the height.