if `x_(1)^(2) +2y_(1)^(2)+3z_(1)^(2)=x_(2)^(2)+2y_(2)^(2)+3z_(2)^(2)=x_(3)^(2)+2y_(3)^(2)+3z_(3)^(2)=2` and `x_(2)x_(3) +2y_(2)y_(3)+3z_(2)z_(3)=x_(3)x_(1)+2y_(3)y_(1)+3z_(3)z_(1)=x_(1)x_(2)+2y_(1)y_(2)+3z_(1)z_(2)=1`
Then find the value of `|{:(x_(1),,y_(1),,z_(1)),(x_(2),,y_(2),,z_(2)),(x_(3),,y_(3),,z_(3)):}|`

Explanation

Let
`Delta= |{:(x_(1),,y_(1),,z_(1)),(x_(2),,y_(2),,z_(2)),(x_(3),,y_(3),,z_(3)):}|`
`:. Delta^(2)= Delta xx Delta`
`=|{:(x_(1),,y_(1),,z_(1)),(x_(2),,y_(2),,z_(2)),(x_(3),,y_(3),,z_(3)):}|xx |{:(x_(1),,y_(1),,z_(1)),(x_(2),,y_(2),,z_(2)),(x_(3),,y_(3),,z_(3)):}|`
` =(1)/(1xx2xx3) |{:(x_(1),,2y_(1),,z_(1)),(x_(2),,2y_(2),,z_(2)),(x_(3),,2y_(3),,z_(3)):}|xx |{:(x_(1),,y_(1),,z_(1)),(x_(2),,y_(2),,z_(2)),(x_(3),,y_(3),,z_(3)):}|`
`= (1)/(6) |{:(x_(1)^(2)+2y_(1)^(2)+3z_(1)^(2),,x_(1)x_(2)+2y_(1)y_(2)+3z_(1)z_(2),,x_(3)x_(1)+2y_(2)y_(3)+3z_(2)z_(3)),(x_(1)x_(2)+2y_(1)y_(2)+3z_(1)z_(2),,x_(2)^(2)+2y_(2)^(2)+3z_(2)^(2),,x_(2)x_(3)+2y_(2)y_(3)+3z_(3)z_(3)),(x_(3)x_(1)+2y_(3)y_(1)+3z_(3)z_(1),,x_(3)^(2)+2y_(3)^(2)+3z_(3)^(2),,):}|`
`=(1)/(6) |{:(2,,1,,1),(1,,2,,1),(1,,1,,2):}|`
`=(2)/(3)`