The three vectors `hat i+hat j,hat j+hat k, hat k+hat i` taken two at a time form three planes, The three unit vectors drawn perpendicular to these planes form a parallelopiped of volume: ________________

Options

A. `1/3`
B. `4`
C. `(3sqrt(3))/4`
D. `4/(3sqrt(3))`

Answer

B

Explanation

Let `veca=hati+hatj, vecb=hatj+hatk, vecc=hatk+hati`
Let `vecn_(1),vecn_(2),vecn_(3)` be the normals to the given planes. Then,
`vecn_(1)=vecaxxvecb, vecn_(2)=vecbxxvecc` and `vecn_(3)=veccxxveca`
`:.` Volume of the given parallelopiped is given by
`[(vecn_(1), vecn_(2), vecn_(3))]=[(veca, vecb, vecc)]^(2)=|(1, 1, 0),(0, 1,1),(1,0, 1)|^(2)=4`