Let `veca=a_(1)hati+a_(2)hatj+a_(3)hatk,vecb=b_(1)hati + b_(2)hatj + b_(3)hatk and vecc=c_(1)hati +c_(2)hatj +c_(3)hatk` be three non-zero vectors such that `vecc` is a unit vector perpendicular to both vectors, `veca and vecb` . If the angle between `veca and vecb 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

Explanation

To solve the problem, we need to find the square of the determinant formed by the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\). Given that \(\vec{c}\) is a unit vector perpendicular to both \(\vec{a}\) and \(\vec{b}\), and the angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{6}\), we can proceed as follows:

### Step 1: Write the vectors
Let:
\[ \vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \]
\[
...