Question

How do you 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)?

Answer 1

The answer is: `V=16`.

Given three vectors, there is a product, called scalar triple product, that gives (the absolute value of it), the volume of the parallelepiped that has the three vectors as dimensions.

So:

`vec(PQ)=(3+1,0-2,1-5)=(4,-2,-4)`

`vec(PR)=(3-5,0-1,1+1)=(-2,-1,2)`

`vec(PS)=(3-0,0-4,1-2)=(3,-4,-1)`

The scalar triple product is given by the determinant of the matrix `(3xx3)` that has in the rows the three components of the three vectors:

`|+4 -2 -4|`
`|-2 -1 +2|`
`|+3 -4 -1|`

and the derminant is given for example with the Laplace rule (choosing the first row):

`4*[(-1)(-1)-(2)(-4)]-(-2)[(-2)(-1)-(2)*(3)+(-4)[(-2)(-4)-(-1)(3)]=`.

`=4(1+8)+2(2-6)-4(8+3)=36-8-44=-16`

So the volume is: `V= |-16|=16`