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Let vec(a)=a(1)hat(i)+a(2)hat(j)+a(3)hat...

Let `vec(a)=a_(1)hat(i)+a_(2)hat(j)+a_(3)hat(k),vec(b)=b_(1)hat(i)+b_(2)hat(j)+b_(3)hat(k)andvec(c)=c_(1)hat(i)+c_(2)hat(j)+c_(3)hat(k)` be three non-zero vectors such that `vec(c)` is a unit vector perpendicular to both `vec(a)andvec(b).` If the angle between `vec(a)andvec(b)" is "(pi)/(6),` then
`{:""|(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3))|"":}^(2)" is equal to"`

A

0

B

1

C

`(1)/(4)(a_(1)^(2)+a_(2)^(2)+a_(3)^(2))(b_(1)^(2)+b_(2)^(2)+b_(3)^(2))`

D

`(3)/(4)(a_(1)^(2)+a_(2)^(2)+a_(3)^(2))(b_(1)^(2)+b_(2)^(2)+b_(3)^(2))(c_(1)^(2)+c_(2)^(2)+c_(3)^(2))`

Text Solution

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The correct Answer is:
C
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Knowledge Check

  • Let overset(to)(a) =a_(1) hat(i) + a_(2) hat(j) + a_(3) hat(k) , overset(to)(a) = b_(1) hat(i) +b_(2) hat(j) +b_(3) hat(k) " and " overset(to)(a) = c_(1) hat(i) +c_(2) hat(j) + c_(3) hat(k) be three non- zero vectors such that overset(to)(c ) is a unit vectors perpendicular to both the vectors overset(to)(c ) and overset(to)(b) . If the angle between overset(to)(a) " and " overset(to)(n) is (pi)/(6) then |{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(c_(1),,c_(2),,c_(3)):}| is equal to

    A
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