The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vectors `hat(a), hat(b), hat(c)` such that `hat(a)*hat(b)=hat(b)*hat(c)=hat(c)*hat(a)=(1)/(2).` Then, the volume of the parallelopiped is

Options

A. `1/(sqrt(2))`
B. `1/(2sqrt(2))`
C. `(sqrt(3))/2`
D. `1/(sqrt(3))`

Answer

A

Explanation

We have
`hata.hata=|hata|^(2)=1, hatb.hatb=1, hatc.hatc=1, hata.hatb=hatb.hatc=hatc.veca=1/2`
`:.[(veca, vecb, vecc)]^(2)=|(hata.hata, hata.hatb, hata.hatc),(hatb.hata,hatb.hatb,hatb.hatc),(hatc.hata,hatc.hatb,hatc.hatc)|`
`implies[(veca, vecb, vecc)]^(2)=|(1,1/2,1/1),(1/2,1,1/2),(1/2,1/2,1)|=1/2`
`implies[(veca, vecb, vecc)]=1/(sqrt(2))` cubic units