Question

Find the volumes of the tetrahedron with vertices $(0,0,0),(1,1,-1),(1,-1,1),(-1,1,1).$

• A. $1/3$
• B. $2/3$
• C. $1$
• D. $4/3$

Answer 1

Answer: (B)

$2/3$

Let $A(0,0,0),B(1,1,-1),C(1,-1,1)$ and $D(-1,1,1)$

$\overrightarrow{AB}=B-A=(1,1,-1)-(0,0,0)=(1,1,-1)\overrightarrow{AC}=(1,-1,1)-(0,0,0)=(1,-1,1)\overrightarrow{AD}=(-1,1,1)-(0,0,0)=(-1,1,1)$

$V=\frac{1}{6}\left|\begin{array}{ccc}1& 1& -1\\ 1& -1& 1\\ -1& 1& 1\end{array}\right|=\frac{1}{6}(1(+1+1)-1(-1-1)-1(1-1))$

$=\frac{1}{6}(2+2-0)=\frac{4}{6}=\frac{2}{3}$