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.
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1Welcome to Mathematics Stack Exchange, I would suggest you to explain a little bit what you tried to solve the problem, so other people could help you better. Good luck! – iadvd May 02 '15 at 14:13
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I only knew how to solve the direction vector of the point C. Didn't get any further. – Eliisa May 02 '15 at 14:26
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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 $$C=A-\frac{l\cdot (A-B)}{(l\cdot l)}l.$$
Alexey Burdin
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