If vector a and b are two non collinear unit vectors and |vector a+ vector b|=√3 then find the value of ( vector a- vector b).(2 vector a+ vector b)

Answer 1

There is probably a more elegant way to solve the problem, but here's one that works.

We know a and b are unit vectors.
By the cosine rule, we can find the angle between the two vectors a+b as:
cos(φ)=(1^2+1^2-(√3)^2)/(2*1*1)=-1/2
=> φ=2π/3.
Without loss of generality (WLOG), we can assume a=<1,0>, from which we determine b=<1/2,(√3)/2>
from there, you can find the dot product as required.