Question

If `a` and `b` are unit vectors and `theta` is the angle between them, express `abs(a-b)` in terms of `theta` ?

Answer 1

`|veca-vecb|=2sin(theta/2).`

Explanation:

Suppose that, `veca, and vecb` are such unit vectors, that,

`hat{((veca, vecb))}=theta, theta in [0,pi].`

`because veca, &, vecb" are unit vectors, ":. |veca|=|vecb|=1....(0).`

We know, `(1): veca*vecb=|veca|*|vecb|*costheta,` and,

`(2): |vecx|^2=vecx*vecx.`

`:., |veca-vecb|^2=(veca-vecb)*(veca-vecb)............[because, (2)],`

`=veca*(veca-vecb)-vecb*(veca-vecb),`

`=veca*veca-veca*vecb-vecb*veca+vecb*vecb,`

`=|veca|^2-2veca*vecb+|vecb|^2...[because, veca*vecb=vecb*veca],`

`=1-2|veca|*|vecb|*costheta+1......[because, (0), and, (1)],`

`=2-2*1*1costheta...........[because, (0)],`

`=2(1-costheta)=2(2sin^2(theta/2)).`

`rArr |veca-vecb|=2sin(theta/2).`

Enjoy Maths.!