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

Explanation

Let `hata` and `hatb` be two unit vectors represented by sides OA and AB of a `DeltaOAB`.

Then, `OA=hata,AB=hatb`
`OB=OA+AB=hata+hatb` (using triangle law of vector addition)
It is given that, `|hata|=|hatb|=|hata+hatb|=1`
`implies|OA|+|AB|=|OB|=1`
`DeltaOAB` is equilateral triangle.
Since, `|OA|=|hata|=1=|=|-hatb|=|AB'|`
therefore, `DeltaOAB'` is a isosceles triangle.
`implies angleAB'O=angleAOB'=30^(@)`
`implies angle BOB'=angleBOA+angleAOB'=60^(@)+30^(@)=90^(@)` (since, `DeltaBOB'` is right angled)
`therefore`In `DeltaBOB'`, we have
`|BB'|=|OB|^(2)+|OB'|^(2)`
`=|hata+hatb|^(2)+|hata-hatb|^(2)`
`2^(2)=1^(2)+|hata-hatb|^(2)`
`hata-hatb|=sqrt(3)` . . . hence proved.