# Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. P(3, 0

Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. $$P(3, 0, 1), Q(-1, 2, 5), R(5, 1, -1), S(0, 4, 2)$$

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$$P(3, 0, 1), Q(-1, 2, 5), R(5, 1, -1), S(0, 4, 2)$$
Find the vectors representing the 3 adjacent edges.
$$PQ=<-1-3,2-0,5-1>=<-4,2,4>$$
$$PR=<5-3,1-0,-1-1>=<2,1,-2>$$
$$PS=<0-3,4-0,2-1>=<-3,4,1>$$
Volume of a parallelpiped
$$\displaystyle{V}={\left|{P}{Q}\cdot{\left({P}{R}\times{P}{S}\right)}\right|}$$
Find cross product first
$$\displaystyle{P}{R}\times{P}{S}={<}{1}{\left({1}\right)}-{\left(-{2}\right)}{\left({4}\right)},{\left(-{2}\right)}{\left(-{3}\right)}-{\left({2}\right)}{\left({1}\right)},{2}{\left({4}\right)}-{\left({1}\right)}{\left(-{3}\right)}\ge{<}{9},{4},{11}{>}$$
Now the dor product
$$V=|PQ*<9,4,11>|$$
$$=|-4(9)+2(4)+4(11)|$$
$$=16$$
Result:
$$\displaystyle{16}{u}{n}{i}{t}{s}^{{{3}}}$$