Let `veca=a_(1)hati+a_(2)hatj+a_(3)hatk,vecb=b_(2)hatj+b_(3)hatk and vecc=c_(1)hati+c_(2)hatj+c_(3)hatk` gve three non-zero vectors such that `vecc` is a unit vector perpendicular to both `veca and vecb`. If the angle between `veca and vecb is pi/6`, then prove that `|{:(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3)):}|p=1/4 (a_(1)^(2)+a_(2)^(2)+a_(3)^(2))(b_(1)^(2)+b_(2)^(2)+b_(3)^(2))`

Options

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))`

Answer

c

Explanation

` (veca xx vecb . Vecc)^(2) = |veca|^(2) |vecb|^(2) |vecc|^(2) sin^(2) theta cos^(2) phi`
(` theta` is the angle between `veca and vecb , phi , =0)`
`1/4 (a_(1)^(2)+ a_(2)^(2) +a_(3)^(2) (b_(1)^(2) +b_(2)^(2) + b_(3)^(2))`