Question

The vector ` vec a=""alpha hat i+2 hat j+""beta hat k` lies in the plane of the vectors ` vec b="" hat"i"+ hat j` and ` vec c= hat j+ hat k` and bisects the angle between ` vec b` and ` vec c` . Then which one of the following gives possible values of `alpha"and"beta` ? (1) `alpha=""2,beta=""2` (2) `alpha=""1,beta=""2` (3) `alpha=""2,beta=""1` (4) `alpha=""1,beta=""1`
A. `alpha = 2, beta = 2 `
B. `alpha = 1, beta = 2 `
C. `alpha = 2, beta = 1 `
D. `alpha = 1, beta = 1 `

Answer 1

Correct Answer - D
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`