# 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)?

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:

$\vec{P Q} = \left(3 + 1 , 0 - 2 , 1 - 5\right) = \left(4 , - 2 , - 4\right)$

$\vec{P R} = \left(3 - 5 , 0 - 1 , 1 + 1\right) = \left(- 2 , - 1 , 2\right)$

$\vec{P S} = \left(3 - 0 , 0 - 4 , 1 - 2\right) = \left(3 , - 4 , - 1\right)$

The scalar triple product is given by the determinant of the matrix $\left(3 \times 3\right)$ 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)]=.

$= 4 \left(1 + 8\right) + 2 \left(2 - 6\right) - 4 \left(8 + 3\right) = 36 - 8 - 44 = - 16$

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