A unit vector in xy-plane that makes an angle of `45^0` with the vector `hati+hatj` and angle of `60^0` with the vector `3hati-4hatj` is (A) `hati` (B) `(hati+hatj)/sqrt(2)` (C) `(hati-hatj)/sqrt(2)` (D) none of these

Options

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

Answer

D

Explanation

Let the required unit vector be `r = ahati + bhatj`
Then, `|r| = 1`
`rArr a^(2) + b^(2) =1" "…(i)`
Since, r makes an angle of `45^(@)` with `hati + hatj` and an angle of `60^(@)` with `3hati - 4hatj`, therefore
`cos""(pi)/(4)*(r*(hati+hatj))/(|r||hati+hatj|)`

and `cos ""(pi)/(3) = (r*(3hati -4hatj))/(|r||3hati-4hatj|)`
`rArr (1)/(sqrt(2)) = (a+b)/(sqrt(2))`
and `(1)/(2) = (3a-4b)/(5)`
`rArr a + b = 1`
and `3a -4b = (5)/(2)`
`rArr a= (13)/(14),b = (1)/(14)`
`therefore r = (13)/(14)hati + (1)/(14)hatj`