Let `overset(to)(a_(r))=x_(r )hat(i)+y_(r )hat(j)+z_(r )hat(k),r=1,2,3` three mutually prependicular unit vectors then the value of `|{:(x_(1),,x_(2),,x_(3)),(y_(1),,y_(2) ,,y_(3)),(z_(1),,z_(2),,z_(3)):}|` is equal to

Options

A. zero
B. `+-1`
C. `+-2`
D. none of these

Answer

B

Explanation

`Delta =|{:(x_(1),,x_(2),,x_(3)),(y_(1),,y_(2),,y_(3)),(z_(1),,z_(2),,z_(3)):}|=|{:(x_(1),,y_(1),,z_(1)),(x_(2),,y_(2),,z_(2)),(x_(3),,y_(3),,z_(3)):}|`
`:. Delta^(2) = |{:(x_(1),,y_(1),,z_(2)),(x_(2),,y_(2),,z_(2)),(x_(3),,y_(3),,z_(3)):}||{:(x_(1),,y_(1),,z_(1)),(x_(2),,y_(2),,z_(2)),(x_(3),,y_(3),,z_(3)):}|`
`=|{:(x_(1)^(2)+y_(1)^(2)+z_(1)^(2),,x_(1)x_(2)+y_(1)y_(2)+z_(1)z_(2),,x_(1)x_(3)+y_(1)y_(3)+z_(1)z_(3)),(x_(1)x_(2)+y_(2)y_(1)+z_(2)z_(1),,x_(2)^(2)+y_(2)^(2)+z_(2)^(2),,x_(2)x_(3)+y_(2)y_(3)+z_(2)z_(3)),(x_(3)x_(1)+y_(3)y_(1)+z_(3)z_(1),,x_(2)x_(3)+y_(2)y_(3)+z_(2)z_(3),,x_(3)^(2)+y_(3)^(2)+z_(3)^(2)):}|`
`=|{:(1,,0,,0),(0,,1,,0),(0,,0,,1):}|=1 rArr Delta =+- 1`