Let `hat u= u_1 hat i+u_2 hat j+u_3 hat k` be a unit vector in be a unit vector in `RR^3 and hat w=1/sqrt6 (hat i+hat j+2 hat k)`.Given that there exists vector `hat v` in `RR^3` such that `|hat u xx vec v|=1 and hat w . (hat u xx vec v)=1`. Which of the following statement(s) is(are) correct?

Options

A. There is exactly one choice for such V
B. There are infinitely many choice for such v
C. if `hat(u)` lines in the XY- plane then `|u_(1)|=|u_(2)|`
D. If `hat(u)` lies in the XY- plane then `2|u_(1)|=|u_(2)|`

Answer

B, C

Explanation

Let 0 be the angle between `hat(u) " and " vec(v)`
`:. |U xx vec(v) |=1 rArr |u| vec(v)|sin 0 =1`
`:. |vec(v)| sin 0 =1`

Clearly there may be infinite vectors `vec(OP) = vec(v)` such that P is always 1 unit distance from `hat(u)`
` :. ` Option (b) is correct .
Again let `phi` be the angle between w w and `uxx vec(v).`
`:. w.(vec(uxx vec(v)) =1 rArr |w| |uxx vec(v)|cos phi =1`
`rArr cos phi =1 rArr phi =0`
Thus ` w=u xx vec(v)`
Now if `hat(u)` lines in XY-plane then
`uxx vec(v) = |{:(hat(i) ,,hat(j),,j),(u_(1),,u_(2),,0),(v_(1),,v_(2),,v_(3)):}|" or " u= u hat(i) + u_(2) hat(j)`
`:. w= (u_(2) v_(3)) hat(i) -(u_(1)v_(3)) hat(j) + (u_(1) u_(2) - v_(1)u_(2))hat(k)`
`= (1)/(sqrt(6)) (hat(i)+hat(j) +2hat(k))`
`:. u_(2) v_(3) = (1)/(sqrt(6)) , u_(1) u_(3) = (-1)/(sqrt(6))`
` rArr (u_(2)u_(3))/(u_(1)u_(3)) =- 1 " or " |u_(1)|=|u_(2)|`
`:.` Option (c ) is correct
Now if `hat(u)` lies in XZ-plane then `u=u_(1) hat(i) + u_(3) hat(k)`
`:. uxx vec(v) = |{:(hat(i) ,,hat(j) ,,hat(k) ),(u_(1),,0,,u_(3)),(v_(1),,v_(2),,v_(3)):}|`
` rArr w = (- v_(2) u_(3)) hat(i) -(u_(1)v_(3) - u_(3)v_(1)) hat(j) + (u_(1) v_(2) ) hat(k)`
`rArr hat(k) = (1)/(sqrt(6)) (hat(i) + hat(j) + 2hat(k))`
` rArr - v_(2) u_(3) = (1)/(sqrt(6)) " and " u_(1) v_(2) = (2)/(sqrt(6))`
`:. |u_(2)|=2 |u_(3)|`
`:.` Option (d) is wrong