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)?

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##