If ` hat a , hat b ,a n d hat c` are unit vectors, then `| hat a- hat b|^2+| hat b- hat c|^2+| hat c- hat a|^2` does not exceed `45^0` b. `60^0` c. `cos^(-1)(1//3)` d. `cos^(-1)(2//7)`

Options

A. 4
B. 9
C. 8
D. 6

Answer

b

Explanation

`hata, hatb and hatc ` are unit vectors, now,
` x= |hata - hatb|^(2) = |hatb - hatc|^(2) + |hatc - hata|^(2)`
` = 1/2 ( hata . Hata + hatb.hatb + hatc. Hatc) - 2hata.hatb - 2hata.hatb- 2hatb.hatc - 2hatc.hata`
`= 6-2 ( hata .hatb + hatb. hatc + hatc.hata)`
Also , `|hata + hatb + hatc| ge 0`
` or hata. hata + hatb. hatb + hatc.hatc + 2 ( hat. hatb + hatb . hatc +hatc hatc) ge 0 `
`or 3+2 ( hata.hatb + hatb. hatc + hatc .hata) ge 0 `
` or 2 ( hata. hatb + hatb. hatc + hatc .hata) ge -3`
`or -2 ( hata. hatb + hatb. hatc +hatc.hata) le 3`
` or 6-2 (hata.hatn + hatb. hatc + hatc.hata) le 9 `
From (i) and (ii) ` x le 9`
therefore, x does not exced 9.