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

Options

A. `sqrt(2)`
B. `sqer(3)`
C. `1//sqer(2)`
D. `sqrt(5)`

Answer

B

Explanation

Let `hat(n)_(1)` and `hat(n)_(2)` be 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 implies cos theta= -1/2 :. theta= 120^(@)`
Now the difference Vector is `hat(n)_(d)=hat(n)_(1)-hat(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= 3 impliesn_(d)= sqrt(3)`