Statement 1: If `a_(1)hati + a_(2)hatj + a_(3)hatk, vecb_(1)hati+b_(2)hatj + b_(3) hatk and c_(1)hati + c_(2)hatj + c_(3)hatk` are three mutually perpendicular unit vectors then `a_(1)hati + b_(1)hatj + c_(1)hatk,a_(2)hati +b_(2)hatj+c_(2) hatk and a_(3)hati + b_(3) hatj + c_(3) hatk` may be mutually perpendicular unit vectors.
Statement 2 : value of determinant and its transpose are the same.

Options

A. Both the statements are true and statement 2 is the correct explanation for statement 1.
B. Both statements are true but statement 2 is not the correct explanation for statement 1.
C. Statement 1 is true and Statement 2 is false
D. Statement 1 is false and Statement 2 is true.

Answer

A

Explanation

Let the three given unit vectors be `hata, hatb and hatc` since they are mutually perpendicular , `hata. (hatb xx hatc) =1 `
therefore,
`|{:(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3)):}|=1`
`|{:(a_(1),b_(1),c_(1)),(a_(2),b_(2),c_(2)),(a_(3),b_(3),c_(3)):}|=1`
Hence, `a_(1)hati +b_(1)hatj + c_(1)hatk , a_(2)hati+b_(2)hatj +c_(2)hatk and a_(3)hati +b_(3)hatj +c_(3)hatk` may be mutually perpendicular.