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))`

Explanation

`|{:(a_(1) , a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3)):}|^(2)= [veca vecb vecc]^(2)`
` = ((veca xx vecb) .vecc)^(2)`
` ( ab sin theta vecc. vecc)^(2)` .
`(a^(2)b^(2))/4`
` 1/4 (a_(1)^(2) + a_(2)^(2)+a_(3)^(2))(b_(1)^(2) =b_(2)^(2) + b_(3)^(2)) `