Question

if a(cap) and b(cap)are unit vectors at an angle theta show that |a(cap)-b(cap)=2(sin theta/2 )

Answer 1

if a(cap) and b(cap) are unit vectors at an angle theta show that $|a(\text{cap}) - b(\text{cap})| = 2$ (sin theta/2)

$\vec{a}$ and $\vec{b}$ - are unit vectors

$\theta$ - the angle between them

Law of cosines:

$\left|\vec{a} -\vec{b}\right|^2 = |\vec{a} |^2 +\left|\vec{b}\right|^2 -2|\vec{a} |\left|\vec{b}\right|\cos \theta ,$

$\vec{a}, \vec{b}$ - some vectors, $\theta$ - the angle between them

$\vec{a}$ and $\vec{b}$ - are unit vectors, therefore $|\vec{a}| = |\vec{b}| = 1$

Law of cosines:

$\left|\vec{a} -\vec{b}\right|^2 = 1 + 1 - 2\cos \theta = 2(1 - \cos \theta) = \left|\sqrt{\frac{1 - \cos\theta}{2}} = \sin \frac{\theta}{2}\right| = 4\sin^2\frac{\theta}{2}$

Or:

$\left|\vec{a} -\vec{b}\right| = 2\sin \frac{\theta}{2}$