Let ` vecu = u_(1)hati + u_(2)hatj +u_(3)hatk` be a unit vector in ` R^(3) and vecw = 1/sqrt6 ( hati + hatj + 2hatk)` , Given that there exists a vector `vecv " in " R^(3)` such that ` | vecu xx vecv| =1 and vecw . ( vecu xx vecv) =1` which of the following statements is correct ?

Options

A. There is exactly one choice for such ` vecv`
B. There are exactly two for such `vecv`
C. There are exactly for such `vecv`
D. There are infinitely many choices for such `vecv`

Answer

D

Explanation

Clearly, `vecw` is a unit vector such that ` |vecu xx vecv|=1` and `vecw` , `(vecu xx vecv) =1 ` Now, `vecw` .`(vecu xx vecv)` =1`
` `Rightarrow vecw` = `vecu` `xx` `vecv`
` Rightarrow |vecw| = |vecu xx vecv|`
` Rightarrow 1 = |vecu||vecv| sin theta " where " theta` is the angle betweeen ` vecu and vecv`
` Rightarrow vecc sin theta =1`
Clearly, p can take infinitely many positions on the line at a unit distance from OA. Consequently, ` vec(OP) =vecv` has, infinitely, many chocies.