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 $A$.
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1$\begingroup$ Give us an attempt or a little idea, so that we can understand your difficulties. $\endgroup$Duca_Conte– Duca_Conte2019-10-08 19:51:31 +00:00Commented Oct 8, 2019 at 19:51
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$\begingroup$ What do you mean by “using vectors?” Off the top of my head, I can think of two very different methods that both use vectors. $\endgroup$amd– amd2019-10-08 22:02:01 +00:00Commented Oct 8, 2019 at 22:02
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2 Answers
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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.
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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.
