A vector → A is rotated by a small angle ∆ θ radians (∆ θ ≪ 1) to...

$\left|\overset{\to}{B}\right| = \left|\overset{\to}{A}\right|$

\left|\overset{\to}{B} - \overset{\to}{A}\right| = \sqrt{{\left|\overset{\to}{A}\right|}^{2} + {\left|\overset{\to}{A}\right|}^{2} + 2\left|\overset{\to}{A}\right|\left|\overset{\to}{A}\right|\cos \left(180 ° - ∆ \theta\right)}$</p><p>$= \sqrt{{\left|\overset{\to}{A}\right|}^{2} + {\left|\overset{\to}{A}\right|}^{2} - 2{\left|\overset{\to}{A}\right|}^{2}cos ∆ \theta}$</p><p>$= \sqrt{2{\left|\overset{\to}{A}\right|}^{2} - 2{\left|\overset{\to}{A}\right|}^{2}cos ∆ \theta}$</p><p>$= \sqrt{2{\left|\overset{\to}{A}\right|}^{2}\left(1 - cos ∆ \theta\right)}$</p><p>$∵ 1 - cos ∆ \theta = 2\sin^{2}\left(\dfrac{∆ \theta}{2}\right)$</p><p>$= \sqrt{2{\left|\overset{\to}{A}\right|}^{2} \times 2\sin^{2}\left(\dfrac{∆ \theta}{2}\right)}$</p><p>$= 2\left|\overset{\to}{A}\right|\sin \left(\dfrac{∆ \theta}{2}\right)$</p><p>For small$∆ \theta, \sin \left(\dfrac{∆ \theta}{2}\right) ≃ \left(\dfrac{∆ \theta}{2}\right)$</p><p>\left|\overset{\to}{B} - \overset{\to}{A}\right| = 2\left|\overset{\to}{A}\right|\left(\dfrac{∆ \theta}{2}\right)$

$= \left|\overset{\to}{A}\right| ∆ \theta$