Question

Find the volume of the parallelepiped with the given vertices:(1,2,1), (2,3,4), (3,4,5), (2,3,2), (3,3,3), (4,4,6), (4,4,4), (5,5,7)

Answer 1

0

Explanation:

three vectors are

`u=(2,3,4)-(1,2,1)=(1,1,3)`

`v=(3,4,5)-(1,2,1)=(2,2,4)`

`w=(2,3,2)-(1,2,1)=(1,1,1)`

volume is given by
`V=u \cdot(v * w)`
`v * w=\left|\begin{array}{ccc}i & j & k \\ 2 & 2 & 4 \\ 1 & 1 & 1\end{array}\right|=(-2,2,0)`

so volume is
`V=(1,1,3) \cdot(-2,2,0)=0`

2

three vectors are
`u=(4,4,6)-(3,3,3)=(1,1,3)`

`v=(4,4,4)-(3,3,3)=(1,1,1)`

`w=(5,5,7)-(3,3,3)=(2,2,4)`

volume is given by
`V=u \cdot(v * w)`
`v * w=\left|\begin{array}{ccc}i & j & k \\ 1 & 1 & 1 \\ 2 & 2 & 4\end{array}\right|=(2,-2,0)`

so volume is
`V=(1,1,3) \cdot(-2,2,0)=0`