The three vectors `hati+hatj, hatj+hatk, hatk+hati` taken two at a time form three planes. The three unit vectors drawn perpendicular to these three planes form a parallelopiped of volume.

Options

A. 43468
B. 4
C. `(3 sqrt(3)) 4`
D. `4//(3sqrt(3))`

Answer

C

Explanation

`Let [{:(veca=hati+hatj),(vecb=hatj+hatk),(vecc=hatk+hati):}`
`hatr_(1)=((vecaxxvecb))/(|vecaxxvecb|)=((hati-hatj+hatk))/(sqrt(3))`
`Similarly , hatr_(2)=((vecbxxvecc))/(|vecbxxvecc|)=(1)/(sqrt(3))(hati+hatj-hatk)`
` hatr_(3)=((vecbxxvecc))/(|vecbxxvecc|)=(1)/(sqrt(3))(-hati+hatj+hatk)`
then , the required
Volume is `impliesV=(1)/(3sqrt(3))|{:(1,-1,1),(1,1,-1),(-1,1,1):}|=(4)/(3sqrt(3))`