If the sum of two unit vectors is a unit vector,then find the magnitude of their differences.

Explanation

Let `hat(n)_(1)` and `hat(n)_(2)` are the two unit vector. Then the sum is
`vec(n)_(s)=hat(n)_(1)+hat(n)_(2)` or `n_(s)^(2)=n_(1)^(2)+n_(2)^(2)+2n_(1)n_(2) cos theta`
`1+1+2 cos theta`
Since it is given that `n_(s)` is also a unit vector,therefore
`1=1+1+2 cos theta`
or `cos theta =-1/2 or theta=120^(@)`
now the diffrence vector is `n_(d)=n_(1)-n_(2)`
or `n_(d)^(2)=n_(1)^(2)+n_(2)^(2)-2n_(1)n_(2) cos theta=1+1-2 cos (120^(@))`
`:. n_(d)^(2)=2-2(-1//2)=2+1=3rArrn_(d)=sqrt(3)`