Hello, I can not solve problem from analytical geometry. If you can help me to find out what is wrong in my procedure and show me a better one I would be really gratefull.

Question

There are given 3 points A(2,2,3), B (6,3,0), C(3,-1,-1). On the x ax-axis lies point X, find coordinates of that point if the volume of a tetrahedron that those points make is 26.

Thank you for your help.

Answer 1

put the coordinates in columns of the matrix M:

(2 6 3 x)
(2 3 -1 0)
(3 0 -1 0)
(1 1 1 1)

The volume of the tetrahedron is

1/3! |M|
= 1/6 (13x-39)
So, you want

1/6 (13x-39) = 26
13x-29 = 156
13x = 185
x = 14.23

google tetrahedron vertices volume and you can find several useful discussions. Heck, maybe even in your own textbook!

Answer 2

That is exactly what I did get but in results it is writen that it has two solutions X(15,0,0) and X (-9,0,0).

That is why I thought my procedure was wrong.
And I have one really similar problem and as well by this procedure I did not get the right two solutions, not even two of them as it is in results.

Answer 3

Hmmm. I can see that there could be two solutions -- one in front of the plane of the given points, and another behind it. But since the matrix has a linear determinant, I'm not sure how to get the other one. Need to think on that a bit.