Science:Math Exam Resources/Courses/MATH152/April 2012/Question 01 (e)/Solution 1

The parallelepiped formed by the vectors $AB$, $AC$ and $AP$ is given by (the absolute value of) the determinant of the three vectors. From 1(a) and 1(c), we have that

$\begin{array}{rl}\overrightarrow{AB}& =[-2,2,1]\\ \overrightarrow{AC}& =[-1,1,1]\end{array}$

and so all we need is $\overrightarrow{AP}$,

$\overrightarrow{AP}=P-A=[0,-1,0].$

Therefore, the volume of the parallelepiped is given by

$\begin{array}{rl}\text{volume}& =\left|\text{det}\left(\begin{array}{ccc}-2& 2& 1\\ -1& 1& 1\\ 0& -1& 0\end{array}\right)\right|\\ & =|-1\text{det}\left(\begin{array}{cc}-2& 1\\ -1& 1\end{array}\right)|\\ & =|-2+1|=|-1|=1.\end{array}$