\(P(3, 0, 1), Q(-1, 2, 5), R(5, 1, -1), S(0, 4, 2)\)

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:

\(\displaystyle{16}{u}{n}{i}{t}{s}^{{{3}}}\)