Find the volume of the parallelepiped with one vertex at A(-1,-4,6) and adjacent vertices at B(-1,-5,4) C(5,-11,-2) D(3,-2,9)

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If you sort of "anchor" the problem at vertex A, then you can think of the three vectors extending from A as vectors `vec(a), vec(b), vec(c)`

That is:

`vec(AB) = vec(a)`

`vec(AC) = vec(b)`

`vec(AD) = vec(c)`

The parallelepiped described by the vectors `vec(a)`, `vec(b)`, and `vec(c)` has volume:

`|vec(a) * ( vec(b) ** vec(c) )|`

So:

`vec(a) = vec(AB) = <-1-(-1), -5-(-4), 4-6> = <0, -1, -2>`

`vec(b) = <6,-7,-8>`

`vec(c) = <4, 2, 3>`

`vec(b)**vec(c) = <(-7*3)-(2*-8),-(6*3)+(-8*4),(6*2)-(-7*4)> = <-5,-50,40>`

`vec(a) * <-5,-50,40> = 0*(-5)+(-1)*(-50)+(-2)*40 = -30

`|vec(a)*(vec(b)**vec(c))| = 30

So the volume of the parallelepiped is 30

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