Question

If `vec"a", vec"b"` are unit vectors and q is the angle between them, show that 

`sin  theta/2 = 1/2|vec"a" - vec"b"|`

Answer 1

`(vec"a" - vec"b")^2 = vec"a"^2 + vec"b"^2 - 2vec"a" * vec"b"`

`|vec"a" - vec"b"|^2 = |vec"a"|^2 + |vec"b"|^2 - 2|vec"a"|*|vec"b"| cos theta`

`|vec"a" - vec"b"|^2 = 1^2 + 1^2 - 2 xx 1*1 cos theta`

= `2 - 2 cos theta`

= `2(1 - cos theta)`

`|vec"a" - vec"b"|^2 = 2 xx 2 sin^2  theta/2`

`|vec"a" - vec"b"|^2 = 2 sin  theta/2`

`sin  theta/2 = 1/2|vec"a" - vec"b"|`