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)?

1 Answer
Massimiliano
Mar 24, 2015

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#

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