If unit vectors A cap and B cap are inclined at an angle θ . . .

Question : If unit vectors Â and B̂ are inclined at an angle θ, then prove that |Â-B̂|=2sinθ/2

Doubt by Ishmeet

Solution : 

|Â-B̂|² = (Â-B̂).(Â-B̂)
|Â-B̂|² = Â.Â-Â.B̂-B̂.Â+B̂.B̂
|Â-B̂|² = |Â|²-Â.B̂-Â.B̂+|B̂|²

|Â-B̂|² = (1)²-2Â.B̂+(1)²
|Â-B̂|² = 1-2Â.B̂+1
|Â-B̂|² = 2-2Â.B̂
|Â-B̂|² = 2-2[ABcosθ]
|Â-B̂|² = 2-2[(1)(1)cosθ]
[∵Magnitude of unit vector is 1]

|Â-B̂|² = 2-2cosθ
|Â-B̂|² = 2[1-cosθ]
[∵2sin²θ=1-cos2θ]
|Â-B̂|² = 2[2sin²θ/2]
|Â-B̂|² = 4sin²θ/2
|Â-B̂| = √[4sin²θ/2]

|Â-B̂| = 2sinθ/2

Hence Proved.