If the sum of two unit vectors is a unit vector, what is the magnitude of the difference of the two vectors?

Let ai+bj and ci+dj be the unit vectors.
From the defination,
#sqrt(a^2+b^2)=1#
#=>(a^2+b^2)=1#__(1)
#sqrt(c^2+d^2)=1#
#=>(c^2+d^2)=1#_____(2)

Add the above two vectors,
(ai+bj)+(ci+dj) =(a+c)i+(b+d)j__(3)

Also given that,
The magnitude of equation(3) is also unity.
#sqrt((a+c)^2+(b+d)^2)=1#
#=>((a+c)^2+(b+d)^2)=1#
#=>(a^2+c^2+2ac)+(b^2+d^2+2bd)=1#
#=>1+2ac+1+2bd=1#
#=>cancel(1)+2ac+2bd+1=cancel(1)#
#=>ac+bd=-1/2#___(4)
the magnitude of their difference is:?
THe difference of two unit vectors is
#(ai+bj)-(ci+dj)=(a-c)i+(b-d)j#
#Magnitude=sqrt((a-c)^2+(b-d)^2#
#=sqrt(a^2+c^2-2ac+b^2+d^2-2bd)#
#=sqrt(1+1-2(ac+bd)# [From (1) and (2)]
#=sqrt(2-2(-1/2)# [From equation(4)]
#=sqrt(2+1)#
#=sqrt(3)#
#=1.73205#

If the sum of two unit vectors is a unit vector, then the magnitude of their difference is 1.73205

For those more visually or geometrically oriented; consider the following diagram, noting that if the sum of two unit vectors is a unit vector then the 3 unit vectors can be combined into an equilateral triangle:

By extending the initial unit vector and dropping a perpendicular to that extension from the terminal point of the vector #vec(a-b)#
we have a right triangle with sides:
#1+1/2# and #sqrt(3)/2#

So
#abs(vec(a-b)) = sqrt((3/2)^2 +(sqrt(3)/2)^2)#

#color(white)("XXXX")##=sqrt((9+3)/4) = sqrt(12/4) = sqrt(3)#