   Chapter 11.4, Problem 39E

Chapter
Section
Textbook Problem

# Volume In Exercises 37 and 38, find the volume of the parallelepiped with the given vertices. ( 0 , 0 , 0 ) , ( 3 , 0 , 0 ) , ( 0 , 5 , 1 ) , ( 2 , 0 , 5 ) , ( 3 , 5 , 1 ) , ( 5 , 0 , 5 ) , ( 2 , 5 , 6 ) , ( 5 , 5 , 6 )

To determine

To calculate: The volume of a parallelepiped with the given vertices,

(0,0,0),(3,0,0)(0,5,1)(2,0,5),(3,5,1),(5,0,5),(2,5,6)(5,5,6).

Explanation

Given:

The vertices are,

(0,0,0),(3,0,0)(0,5,1)(2,0,5),(3,5,1),(5,0,5),(2,5,6)(5,5,6)

Formula used:

For the vectors given as, u=a1i+b1j+c1k,v=a2i+b2j+c2k and w=a3i+b3j+c3k,

u(v×w)=|a1b1c1a2b2c2a3b3c3|

Calculation:

Let the vertices of the given parallelepiped be with reference to the origin:

OA=(0,0,0)OB=(3,0,0)OC=(0,5,1)OD=(2,0,5)

And,

OE=(3,5,1)OF=(5,0,5)OG=(2,5,6)OH=(5,5,6)

The vectors with the same initial point are calculated as:

AB=OBOA=<3,0,0><0,0,0>=<3,0,0>

AC=OCOA=<0,

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