Let ^ a , ^ b be unit vectors. If → c be a vector such that the...

We have, \overset{^}{b} = \overset{\to}{c} + 2\left(\overset{\to}{c} \times \overset{^}{a}\right),

\overset{^}{b} \cdot \overset{^}{b} = \left(\overset{\to}{c} + 2\left(\overset{\to}{c} \times \overset{^}{a}\right)\right) \cdot \left(\overset{\to}{c} + 2\left(\overset{\to}{c} \times \overset{^}{a}\right)\right)

⇒ 1 = \left(\overset{\to}{c} \cdot \overset{\to}{c}\right) + \overset{\to}{c} \cdot 2\left(\overset{\to}{c} \times \overset{^}{a}\right) + 2\left(\overset{\to}{c} \times \overset{^}{a}\right) \cdot \overset{\to}{c} + 4\left(\overset{\to}{c} \times \overset{^}{a}\right) \cdot \left(\overset{\to}{c} \times \overset{^}{a}\right)

⇒ 1 = {\left|\overset{\to}{c}\right|}^{2} + 0 + 0 + {\left|\overset{\to}{c} \times \overset{^}{a}\right|}^{2} ⇒ 1 = {\left|\overset{\to}{c}\right|}^{2} + 4{\left|\overset{\to}{c}\right|}^{2}{\left|\overset{^}{a}\right|}^{2}\sin^{2}\dfrac{\pi}{12}

$⇒ 1 = {\left|\overset{\to}{c}\right|}^{2} + 4{\left|\overset{\to}{c}\right|}^{2}\left(\dfrac{1 - cos\dfrac{\pi}{6}}{2}\right)$

$⇒ 1 = {\left|\overset{\to}{c}\right|}^{2}\left[1 + 2\left(1 - \dfrac{\sqrt{3}}{2}\right)\right]$ $⇒ 1 = {\left|\overset{\to}{c}\right|}^{2}\left[3 - \sqrt{3}\right]$ $⇒ {\left|\overset{\to}{c}\right|}^{2} = \dfrac{1}{3 - \sqrt{3}} = \dfrac{3 + \sqrt{3}}{6}$

$∴ 36{\left|\overset{\to}{c}\right|}^{2} = \dfrac{3 + \sqrt{3}}{6} \times 36 = 6\left(3 + \sqrt{3}\right)$