If sum of two unit vectors is a unit vector; prove that the magnitude of their difference is `sqrt3`

Options

A. `sqrt2`
B. `sqrt3`
C. `sqrt5`
D. `sqrt7`

Answer

B

Explanation

`OA = hata , Ab = hatb OB = hata + hatb`

Also, `|hata|=|hatb|=|hata+hatb| =1`
So, `Delta OAB` is an equilateral triangle.
`therefore |OA|=|hata| = 1= |-hatb|= |AB'|`
Thus, `Delta OAB' ` is an isosceles triangle.
`angle AB' O= angleAOB'=30^(@)`
`and angle BOB' = angleBOA+angleAOB' = 90^(@)`
`therefore " In " DeltaBOB', |B B'|^(2)=|OB|^(2)+|OB' |^(2)`
`rArr 2^(2) = |hata + hatb|^(2) + |hata-hatb|^(2)`
`rArr |hata - hatb|=sqrt(4-1)=sqrt(3)`