A unit vector `hata` makes an angle `pi/4` with z-axis, `if hata+hati+hatj` is a unit vector then `hata` is equal to (A) `hati+hatj+hatk/2` (B) `hati/2+hatj/2-hatk/sqrt(2)` (C) `-hati/2-hat/2+hatk/sqrt(2)` (D) `hati/2-hatj/2-hatk/sqrt(2)`

Options

A. `(hati)/(2)+(hatj)/(2)+(hatk)/(sqrt(2))`
B. `(hati)/(2)+(hatj)/(2)-(hatk)/(sqrt(2))`
C. `-(hati)/(2)-(hatj)/(2)+(hatk)/(sqrt(2))`
D. none of these

Answer

C

Explanation

Let `a=l hati+m hatj+n hatk`, where `l^(2)+m^(2)+n^(2)=1`. A makes an angle `(pi)/(4)` with Z-axis.
`therefore n=(1)/(sqrt(2)),l^(2)+m^(2)=(1)/(2)` . . . (i)
`therefore a=l hati+mhatj+(hatk)/(sqrt(2))`
`a+hati+hatj=(l+1)hati+(m+1)hatj+(hatk)/(sqrt(2))`
Its magnitude is 1, hence `(l+1)^(2)+(m+1)^(2)=(1)/(2)` . . . (ii)
From eqs. (i) and (ii), we get
`2lm=(1)/(2)impliesl=m=-(1)/(2)`
hence, `a=-(hati)/(2)-(hatj)/(2)+(hatk)/(sqrt(2))`.