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

Anonym 2021-09-28 Answered

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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Expert Answer

SabadisO
Answered 2021-09-29 Author has 26210 answers

\(P(3, 0, 1), Q(-1, 2, 5), R(5, 1, -1), S(0, 4, 2)\)
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:
\(\displaystyle{16}{u}{n}{i}{t}{s}^{{{3}}}\)

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