Let u ^ = u 1 i ^ + u 2 j ^ + u 3 k ^ be a unit vector in R 3 and w ^ = 1 6 i ^ + j ^ + 2 k ^ . Given that there exists a vector v → in R 3 such that u ^ × v → = 1 and w ^ . u ^ × v → = 1 . Which of the following statement(s) is (are) correct ?

Step-by-step solution

As | w ^ | = | u ^ | = 1 = | u ^ × v → | Let ϕ is angle between w ^ & ( u ^ × v → ) w ^ u ^ × v → cos ⁡ ϕ = 1 ⇒ ϕ = 0 ⇒ u ^ × v → = w ^ ⇒ There may be infinite vectors v → Satisfying this condition If u ^ his in x y plane : u ^ × v → = i ^ j ^ k ^ u 1 u 2 0 v 1 v 2 v 3 = w ^ ⇒ u 3 = 0 Let v → = v 1 i ^ + v 2 j ^ + v 3 k ^ w ^ = u 2 v 3 i ^ - u 1 v 3 j ^ + u 1 v 2 - u 2 v 1 k ^ = 1 6 i ^ + 1 6 j ^ + 2 6 k ^ u 2 v 3 = 1 6 , - u 1 v 3 = 1 6 ⇒ u 1 = u 2 If u ^ his in x z plane ⇒ u 2 = 0 : u ^ × v → = i ^ j ^ k ^ u 1 0