What are the values of a and b if vectors `a hati+b hatj = hatn and (hati+hatj)` are perpendicular to each other ?

Options

A. 1,0
B. 0,-1
C. 2,1
D. `((1)/(sqrt(2))), (-(1)/(sqrt(2)))`

Answer

D

Explanation

For `a hati + bhatj = hatn," we have "sqrt (a^(2) +b^(2))=1`
`therefore a^(2) + b^(2) =1" "therefore (hatn" is a unit vector")" …(1)"` and since the two vectors are perpendicular
`(a hati + b hatj ) cdot ( hati + hatj) =0`
`therefore a+b =0" "therefore b=-a`
From (1), `a^(2) + (-a)^(2)=1`
`therefore 2a^(2) =1" " therefore a=(1)/(sqrt(2)) and b =-(1)/(sqrt(2))`