Question

If ` hat a` and ` hat b` are unit vectors inclined at an angle `theta` , then prove that `costheta/2=1/2| hat a+ hat b|` `tantheta/2=1/2|( hat a- hat b)/( hat a+ hat b)|`

Answer 1

Here, `veca` and `vecb` are unit vectors.
`:. |hata| = |hatb| = 1`
(i) `|hata+hatb|^2 = (hata)^2+(hatb)^2+2hata*hatb`
`=|hata|^2+|hatb|^2+2|hata||hatb|cos theta`
`=1^2+1^2+2(1)(1)costheta`
`=>|hata+hatb|^2=2(1+costheta)`
`=>|hata+hatb|^2/2 = 1+costheta`
`=>|hata+hatb|^2/2 = 2cos^2(theta/2)`
`=>|hata+hatb|^2/4 = cos^2(theta/2)`
`=>cos (theta/2) =1/2 |hata+hatb| ->(1)`

(ii) `|hata-hatb|^2 = (hata)^2+(hatb)^2-2hata*hatb`
`=|hata|^2+|hatb|^2-2|hata||hatb|cos theta`
`=1^2+1^2-2(1)(1)costheta`
`=>|hata-hatb|^2=2(1-costheta)`
`=>|hata-hatb|^2/2 = 1-costheta`
`=>|hata-hatb|^2/2 = 2sin^2(theta/2)`
`=>|hata-hatb|^2/4 = sin^2(theta/2)`
`=>sin (theta/2) =1/2 |hata-hatb| ->(2)`
Now, dividing (2) by (1),
`sin (theta/2)/cos (theta/2) = |hata-hatb|/|hata+hatb|`
`=> tan (theta/2) = |hata-hatb|/|hata+hatb|`