QJEEM22Ma71f8: JEE Main 2022 Physics - Class 11 JEEM22Ma71f8 Solution: Let and be two unit vectors such that . If (0, ) is the angle…

$\left| {\widehat a + \widehat b + 2(\widehat a \times \widehat b)} \right| = 2,\,\theta \in (0,\,\pi )$

$\Rightarrow {\left| {\widehat a + \widehat b + 2(\widehat a \times \widehat b)} \right|^2} = 4$

$\Rightarrow {\left| {\widehat a} \right|^2} + {\left| {\widehat b} \right|^2} + 4{\left| {\widehat a \times \widehat b} \right|^2} + 2\widehat a\,.\,\widehat b = 4$

$\therefore$ $\cos \theta = \cos 2\theta$

$\therefore$ $\theta = {{2\pi } \over 3}$

where $\theta$ is angle between $\widehat a$ and $\widehat b$ .

$\therefore$ $2\left| {\widehat a \times \widehat b} \right| = \sqrt 3 = \left| {\widehat a - \widehat b} \right|$

(S1) is correct.

And projection of $\widehat a$ on $(\widehat a + \widehat b) = \left| {{{\widehat a\,.\,(\widehat a + \widehat b)} \over {\left| {\widehat a + \widehat b} \right|}}} \right| = {1 \over 2}$

(S2) is correct.