Vectors: right triangle, two vertex known and a direction vector parallel to unknown point

Question

The endpoints of the hypotenuse of a right triangle ABC are A(-10,10,9) and B(14,0,-4). The point C lies on the line that passes through the point A and is parallel to the vector 2i-2j-k. Determine the coordinates of the point C.

Answer 1

The line: $X=A+tl$, $l=(2,-2,-1)$. $C$ is on the line, so $C=A+tl$.
$\angle ACB = \frac{\pi}{2}$, so $(CA\cdot CB)=0$.
$$(A+tl-A)\cdot (A+tl-B) = 0$$ $$(tl)\cdot (A-B) + (tl)\cdot (tl) =0$$ $$l\cdot (A-B) =- t(l\cdot l)$$ $$t=-\frac{l\cdot (A-B)}{(l\cdot l)}$$ And then plug it back to $C=A+tl$ so $$C=A-\frac{l\cdot (A-B)}{(l\cdot l)}l.$$