Let $\widehat a$ and $\widehat b$ be two unit vectors such that $|(\widehat a + \widehat b) + 2(\widehat a \times \widehat b)| = 2$. If $\theta$ $\in$ (0, $\pi$) is the angle between $\widehat a$ and $\widehat b$, then among the statements :

(S1) : $2|\widehat a \times \widehat b| = |\widehat a - \widehat b|$

(S2) : The projection of $\widehat a$ on ($\widehat a$ + $\widehat b$) is ${1 \over 2}$

<p>$$\left| {\widehat a + \widehat b + 2(\widehat a \times \widehat b)} \right| = 2,\,\theta \in (0,\,\pi )$$</p> <p>$$ \Rightarrow {\left| {\widehat a + \widehat b + 2(\widehat a \times \widehat b)} \right|^2} = 4$$</p> <p>$$ \Rightarrow {\left| {\widehat a} \right|^2} + {\left| {\widehat b} \right|^2} + 4{\left| {\widehat a \times \widehat b} \right|^2} + 2\widehat a\,.\,\widehat b = 4$$</p> <p>$\therefore$ $\cos \theta = \cos 2\theta$</p> <p>$\therefore$ $\theta = {{2\pi } \over 3}$</p> <p>where $\theta$ is angle between $\widehat a$ and $\widehat b$.</p> <p>$\therefore$ $$2\left| {\widehat a \times \widehat b} \right| = \sqrt 3 = \left| {\widehat a - \widehat b} \right|$$</p> <p>(S1) is correct.</p> <p>And projection of $\widehat a$ on $$(\widehat a + \widehat b) = \left| {{{\widehat a\,.\,(\widehat a + \widehat b)} \over {\left| {\widehat a + \widehat b} \right|}}} \right| = {1 \over 2}$$</p> <p>(S2) is correct.</p>