Question

The volume of the tetrahedron formed by the points $(1,1,1),(2,1,3),(3,2,2)$ and $(3,3,4)$ in cubic units is

• A. 5/6
• B. 6/5
• C. 5
• D. 2/3

Answer 1

Answer: (A)

Let $A(1,1,1),B(2,1,3),C(3,2,2)$ and $D(3,3,4)$
$\overline{AB}=(2-1,1-1,3-1)=(1,0,2)$
$\overline{AC}=(3-1,2-1,2-1)=(2,1,1)$
$\overline{AD}=(3-1,3-1,4-1)=(2,2,3)$
$V=\frac{1}{6}|[\bar{u},\bar{v},\bar{w}]|$
$[\bar{u},\bar{v},\bar{w}]\left|\begin{array}{ccc}1& 0& 2\\ 2& 1& 1\\ 2& 2& 3\end{array}\right|=1\left(3-2\right)+2\left(4-2\right)$
$=1+2\left(2\right)=5\Rightarrow V=\frac{1}{6}.5=\frac{5}{6}$ cubic units