   Chapter 11.4, Problem 40E

Chapter
Section
Textbook Problem

# Volume In Exercises 37 and 38, find the volume of the parallelepiped with the given vertices. ( 0 , 0 , 0 ) , ( 0 , 4 , 0 ) , ( − 3 , 0 , 0 ) , ( − 1 , 1 , 5 ) , ( − 3 , 4 , 0 ) , ( − 1 , 5 , 5 ) , ( − 4 , 1 , 5 ) , ( − 4 , 5 , 5 )

To determine

To calculate: The volume of the given parallelepiped with the given vertices, (0,0,0),(0,4,0)(3,0,0)(1,1,5),(3,4,0),(1,5,5),(4,1,5)(4,5,5).

Explanation

Given:

The vertices,

(0,0,0),(0,4,0)(3,0,0)(1,1,5),(3,4,0),(1,5,5),(4,1,5)(4,5,5)

Formula used:

The triple scaler product is,

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

u(v×w)=|a1b1c1a2b2c2a3b3c3|

Determinant can be calculated as,

|a1b1c1a2b2c2a3b3c3|=a1(b2c3c2b3)b1(a2c3c2a3)+c1(a2b3b2a3)

Calculation:

Let the vertices of the given parallelepiped are:

OA=(0,0,0)OB=(0,4,0)OC=(3,0,0)OD=(1,1,5)

And,

OE=(3,4,0)OF=(1,5,5)OG=(4,1,5)OH=(4,5,5)

The vectors with the same initial point are calculated as:

AB=OB

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