The vectors `a=alphahat(i)+2hat(j)+betahat(k)` lies in the plane of the vectors `b=hat(i)+hat(j) and c=hat(j)+hat(k)` and bisects the angle between b and c. Then, which one of the following gives possible values of `alpha and beta`?

Options

A. `alpha = 2, beta = 2 `
B. `alpha = 1, beta = 2 `
C. `alpha = 2, beta = 1 `
D. `alpha = 1, beta = 1 `

Answer

D

Explanation

It is given that `veca` besects the angle between `vecb and vec c`. Therefore , `vec a` is parallel `(vec b + vec c )/(2)`
So ` vec a = lambda (hat b + hat c ) `
`rArr alpha hati + 2 hatj + beta hatk = lambda ((hati + hatj ) /( sqrt(2))+(hatj + hatk )/(sqrt(2)))`
`rArr alpha hati + 2 hatj + beta hatk = (lambda) /( sqrt(2)) (hati + 2 hat j + hatk ) `
`rArr alpha = (lambda )/(sqrt(2)) , 2 =sqrt(2) lambda and beta = (lambda )/(sqrt(2)) rArr alpha 1, beta =1`