Unit vector `hat(P)` and `hat(Q)` are inclined at an angle `theta`. Prove that `|hat(P)-hat(Q)|= 2 sin(theta//2)`.

Options

A. `(2 sin)(theta)/2`
B. `(2 cos)(theta)/2`
C. `(2 tan)(theta)/2`
D. `tan theta`

Answer

A

Explanation

`|hat(A)-hat(B)|^(2)= (hat(A)-hat(B)).(hat(A)-hat(B))`
`=hat(A).hat(A)-hat(A).hat(B)-hat(B).hat(A)+hat(B).hat(B)`
`= 1-hat(A).hat(B)-hat(A).hat(B)+1`
`=2-2 cos theta= 2(1-cos theta)`
`=2((2sin)^(2)(theta)/2)= 4 (sin^(2))(theta)/2`
`hat(A)-hat(B)| = (2sin^(2))(theta)/2`