Let ` overset(to)(a) =a_(1) hat(i) + a_(2) hat(j) + a_(3) hat(k) , overset(to)(b) = b_(1) hat(i) +b_(2) hat(j) +b_(3) hat(k) " and " overset(to)(c) = 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)):}|^2` is equal to

Options

A. 0
B. 1
C. `(1)/(4) (a_(1)^(2) +a_(2)^(2) +a_(3)^(2))(b_(1)^(2)+b_(1)^(2) +b_(3)^(2))`
D. `(3)/(4) (a_(1)^(2)+a_(2)^(2))(b_(1)^(2)+b_(2)^(2)+a_(3)^(2))(c_(1)^(2)+c_(2)^(2)+c_(3)^(2))`

Answer

C

Explanation

Since , `(vec(a) xx vec(b)) = | vec(a) ||vec(b)| sin .(pi)/(6) . , hat(n)`
` (vec(a) xx vec(b)) "." vec(c ) = (1)/(2) |vec(a) ||vec(b)|"." vec(n) "." vec(c )`
` [vec(a) vec(b) vec( c)] = (1)/(2) |vec(a) ||vec(b)| "." cos 0^(@)`
`:' hat(n)` is perpendicular to both `vec(a ) , vec(b) , " and " vec(c) ` is also a unit vector perpendicular to both `vec(a) " and " vec(b)`
`:. |{:(a_(1),,a_(2),,a_(3)),(b_(1),,b_(2),,b_(3)),(c_(1),,c_(2),,c_(3)):}|=[vec(a) vec(b) vec( c)]^(2) =(1)/(4) .|vec(a)|^(2)|vec(b)|^(2)`
` = (1)/(4) (a_(1)^(2) +a_(4)^(2) +a_(3)^(2) )(b_(1)^(2)+b_(2)^(2)+b_(3)^(2))`