# Thread: Finding a vertex of a tetrahedron given the volume (in Euclidian space).

1. ## Finding a vertex of a tetrahedron given the volume (in Euclidian space).

So given that a particular pyramid (ABCD) in 3D euclidian space (with a triangular base) has a volume of 6 unitsł, how can you find the coordinates of a point F which lies on the line r= (t-1)i +(-t+1)j + (3t-3)k given that the volume of ABCF is the same as the volume of ABCD.

for the record: A(1,2,3) B(2,3,1) C(3,1,2) D (6,6,6)

help VERY appreciated !! thanks!!

2. Originally Posted by Yehia
So given that a particular pyramid (ABCD) in 3D euclidian space (with a triangular base) has a volume of 6 unitsł, how can you find the coordinates of a point F which lies on the line r= (t-1)i +(-t+1)j + (3t-3)k given that the volume of ABCF is the same as the volume of ABCD.

for the record: A(1,2,3) B(2,3,1) C(3,1,2) D (6,6,6)

help VERY appreciated !! thanks!!
1. The volume of a terahedron with the edges $\displaystyle \vec a, \vec b, \vec c$ is calculated by

$\displaystyle V=\frac16 (\vec a \times \vec b) \cdot \vec c$

$\displaystyle V = \frac16 \left(\overrightarrow{AB} \times \overrightarrow{AC} \right) \cdot \overrightarrow{AF}$
3. Since $\displaystyle \overrightarrow{AF} = [t - 2, -t - 1, 3t - 6]$ you have to solve for t:
$\displaystyle \frac16 ([1, 1, -2] \times [2, -1, -1])\cdot [t - 2, -t - 1, 3t - 6] = 6$