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The edges of a parallelopiped are of uni...

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)/(3)`. Then, the volume of the parallelopiped is :-

A

`(1)/(2)`

B

`(1)/(3)`

C

`(sqrt(3))/(2)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
D
Volume of parallopiped `v^(2)=[hat(a)hat(b)hat(c )]`
`v^(2)=[hat(a)hat(b)hat(c )]^(2)=|(hat(a).hat(a),hat(a).hat(b),hat(a).hat(c )),(hat(b).hat(a),hat(b).hat(b),hat(b).hat(c )),(hat(c ).hat(a),hat(c ).hat(b),hat(c ).hat(c ))|`
`=|(1,1//3,1//3),(1//3,1,1//3),(1//3,1//3,1)|`
`=1(1-1//9)-(1)/(3)((1)/(3)-(1)/(9))+(1)/(3)((1)/(9)-(1)/(3))`
`= (8)/(9)-(1)/(3)((2)/(9))+(1)/(3)((-2)/(9))`
`=(8)/(9)-(4)/(27)`
`v^(2)=(24-4)/(27)=(20)/(27)`
`v = sqrt((20)/(27))`
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