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The edges of a parallelopied are of un...

The edges of a parallelopied are of unit length and are parallel to non- coplanar unit vector `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

A

`(1)/(sqrt(2))` cu unit

B

`(1)/(2sqrt(2))` cu unit

C

`(sqrt(3))/(2) ` cu unit

D

` (1)/(sqrt(3))` cu unit

Text Solution

Verified by Experts

The correct Answer is:
A
The volume of the parallelopied with coterminus edges as `hat(a) , hat(b), hat(c )` is given by `[hat(a), hat(b), hat(c ) ] = hat(a) . (hat(b) xx hat(c ))`

Now `[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//2,,1//2),(1//2,,1,,1//2),(1//2,,1//2,,1):}|`
`rArr [hat(a), hat(b) ,hat(c )]^(2)=1 (1-(1)/(4)) -(1)/(2) ((1)/(2) -(1)/(4)) + (1)/(2) ((1)/(4)-(1)/(2)) = (1)/(2)`
Thus the required volume of the parallelopiped
`=(1)/(sqrt(2)) ` cu unit
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