Coordinates of the altitude

Question

Suppose we have points $P(1,3,2), Q(0,-1,1), R(2,1,0)$. Let's consider a triangle PQR. Let's draw line segment from $R$ which is orthogonal to the side $QP$. Suppose it intersects this line at the point $S$. How to find coordinates of the $S$?

Can anyone help with this problem please.

@HVxvejjw, why it should be a midpoint? General it may not be a mid point. Could you show your reasoning? — RFZ, Oct 08 '19 at 14:28
In an Isoceles triangle the altitude drawn from the vertex bisects the base. — , Oct 08 '19 at 14:30

score 0 · Answer 1 · answered Oct 08 '19 at 14:31

We can write the equation of the $PQ$ line, in particular for a point $S$ as $$S=P+t(P-Q)=(1,3,2)+t[(0,-1,1)-(1,3,2)]=(1,3,2)+t(-1,-4,-1)$$ Write now the condition that $SR$ is perpendicular to $PQ$ using the scalar product: $$(R-S)\cdot(P-Q)=0$$ Plug in the values for $P$, $Q$, $R$, and $S$, and you get an equation for $t$. This is a simple linear equation. Solve it. Then take the answer and use it in the formula for $S$.

Coordinates of the altitude

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