I have tetrahedron of volume 1, with vertices $(1, 2, 3)$ , $(2, 3, 4)$ and $(2, 4, 5)$. The assignment tells me to determine the 4th. It lies on line $(x, y, z) = t(1, 1, 2)$.
I know the volume of a tetrahedron is $ V = \frac{1}{6} determinant|4 (vertices)|$ Since I know this is supposed to be 1, I thought maybe I can set one of the vertices inside to x, y, z and decide what they should be?? Since I know the line it should be on (x, y, z) = t(1, 1, 2) I know I should probably be able to do something with this information too, but what?? I calculated the (x, y, z) with the determinant, atlest tried to and got it to be $(x, y, z) = (0, \frac{1}{6}+z, y-\frac{1}{6})$ Buuut I have no idea if this is somewhat right. So can someone help me. How can I from this information find the 4th vertices of the tetrahedron??
The excercise is in linear algebra.
Hint:
The volume is: $$V=\dfrac{1}{6}\left|\begin{array}{cccc} 1 & 2 & 3 & 1 \\ 2 & 3 & 4 & 1 \\ 2 & 4 & 5 & 1 \\ t & t & 2t &1 \\ \end{array} \right|=1 $$ so you have the equation in $t$ $$\left|\begin{array}{cccc} 1 & 2 & 3 & 1 \\ 2 & 3 & 4 & 1 \\ 2 & 4 & 5 & 1 \\ t & t & 2t &1 \\ \end{array} \right|=6 $$