If the sum of two unit vectors is a unit vector, then the magnitude of their difference is

Options

A. 1
B. 2
C. `sqrt(3)`
D. `2sqrt(3)`

Answer

C

Explanation

Let `hat(a) and hat(b)` be two vectors represented by sides OA and AB of a triangle OAB. Then, `vec(O)A = hat(a), vec(A B)=hat(b) and`
` vec(O) B=vec(O)A+vec(A)B=hat(a)+hat(b) " " ` [By triangle law of add.]

It is given that
`|hat(a)| =|hat(b)|=|hat(a)+hat(b)|=1`
`rArr OA=AB=OB=1`
`rArr Delta OAB` is a equilateral triangle
Since `OA=|hat (a)|=1=|-hat(b)|=vec(A)B' `
`therefore Delta OAB' ` is an isoscles triangle
`rArr angle AB'OP = angle AOB' =30^(@)`
`rArr angle BOB'=angle BOA+angle AOB' = 60^(@)+30^(@)=90^(@)`
In `Delta BOB' ` we have
`BB'^(2) =OB^(2)+OB'^(2)`
`rArr 2^(2) = |hat(a) +hat(b)|^(2)+|hat(a) -hat(b)|^(2)`
`rArr 2^(2)=1^(2)+|hat(a)-hat(b)|^(2) rArr |hat(a)-hat(b)|=sqrt(3)`