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)?

Question

Scarlett Allison · Accepted Answer

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#

Christopher Agresta · Answer

undefined To find the volume of the parallelepiped with adjacent edges $ \overrightarrow{pq} $, $ \overrightarrow{pr} $, and $ \overrightarrow{ps} $, where $ p(3,0,1) $, $ q(-1,2,5) $, $ r(5,1,-1) $, and $ s(0,4,2) $, you can use the scalar triple product formula:

$$ V = | \overrightarrow{pq} \cdot (\overrightarrow{pr} \times \overrightarrow{ps}) | $$

Where $ \overrightarrow{pq} $ is the vector from $ p $ to $ q $, $ \overrightarrow{pr} $ is the vector from $ p $ to $ r $, and $ \overrightarrow{ps} $ is the vector from $ p $ to $ s $.

The cross product $ \overrightarrow{pr} \times \overrightarrow{ps} $ can be calculated using the determinant of a matrix:

$$ \overrightarrow{pr} \times \overrightarrow{ps} = \begin{pmatrix} i & j & k \ x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \ x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \end{pmatrix} $$

Where $ (x_1, y_1, z_1) $, $ (x_2, y_2, z_2) $, and $ (x_3, y_3, z_3) $ are the coordinates of points $ p $, $ r $, and $ s $ respectively.

Once you have the cross product, you can then calculate the dot product with $ \overrightarrow{pq} $, take the absolute value, and find the volume of the parallelepiped.