killjoy1990xb9

2021-12-16

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

Ben Owens

As we have three vectors, there is a product, called scalar triple product, that gives the volume of the parallelepiped that has the three vectors as dimensions.
$P{Q}^{\to }=\left(3+1,0-2,1-5\right)=\left(4,-2,-4\right)$
$P{R}^{\to }=\left(3-5,0-1,1+1\right)=\left(-2,-1,2\right)$
$P{S}^{\to }=\left(3-0,0-4,1-2\right)=\left(3,-4,-1\right)$
The derminant is given for example with the Laplace rule:
$4×\left[\left(-1\right)\left(-1\right)-\left(2\right)\left(-4\right)\right]-\left(-2\right)\left[\left(-2\right)\left(-1\right)-\left(2\right)×\left(3\right)+\left(-4\right)\left[\left(-2\right)\left(-4\right)-\left(-1\right)\left(3\right)\right]=4\left(1+8\right)+2\left(2-6\right)-4\left(8+3\right)=36-8-444=-16$
Thus, $V=16$

rodclassique4r

$PQ=\left(3+1,0-2,1-5\right)=\left(4,-2,-4\right)$
$PR=\left(3-5,0-1,1+1\right)=\left(-2,-1,2\right)$
$PS=\left(3-0,0-4,1-2\right)=\left(3,-4,-1\right)$
Thus, $V=16$ is the volume

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