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

${\begin{aligned}{\vec {AB}}&=[-2,2,1]\\{\vec {AC}}&=[-1,1,1]\end{aligned}}$

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

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

Therefore, the volume of the parallelepiped is given by

${\begin{aligned}{\textrm {volume}}&=\left|{\textrm {det}}{\begin{pmatrix}-2&2&1\\-1&1&1\\0&-1&0\end{pmatrix}}\right|\\&=\left|-1{\textrm {det}}{\begin{pmatrix}-2&1\\-1&1\end{pmatrix}}\right|\\&=|-2+1|=|-1|=1.\end{aligned}}$