Question

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

$\left(S1\right):2|\hat{a}\times \hat{b}|=|\hat{a}-\hat{b}|$
$\left(S2\right):$ The projection of $\hat{a}$ on $(\hat{a}+\hat{b})$ is $\frac{1}{2}$

• A. Only $\left(S1\right)$ is true
• B. Both $\left(S1\right)$ and $\left(S2\right)$ are true
• C. Both $\left(S1\right)$ and $\left(S2\right)$ are false
• D. Only $\left(S2\right)$ is true

Answer 1

The correct option is B Both $\left(S1\right)$ and $\left(S2\right)$ are true

Given, $|\hat{a}+\hat{b}+2(\hat{a}\times \hat{b})|=2$
$\Rightarrow |\hat{a}+\hat{b}+2(\hat{a}\times \hat{b}){|}^2=4$.
$\Rightarrow |\hat{a}{|}^2+|\hat{b}{|}^2+4|\hat{a}\times \hat{b}{|}^2+2\hat{a}\cdot \hat{b}+4\hat{a}\cdot (\hat{a}\times \hat{b})+4\hat{b}\cdot (\hat{a}\times \hat{b})=4.$
$\Rightarrow 1+1+4|\hat{a}\times \hat{b}{|}^2+2\hat{a}\cdot \hat{b}+0+0=4$
$\Rightarrow 2+4{\sin }^2\theta +2\cos \theta =4$
$\Rightarrow 2{\cos }^2\theta -\cos \theta -1=0$
$\Rightarrow (2\cos \theta +1)(\cos \theta -1)=0$
$\Rightarrow \cos \theta =-\frac{1}{2},1$(rejected)
$\Rightarrow \cos \theta =-\frac{1}{2}$
$\therefore \theta =\frac{2\pi }{3}$

Now, $\left(S1\right):$
$2|\hat{a}\times \hat{b}|=2\sin \left(\frac{2\pi }{3}\right)=\sqrt{3}$
and $|\hat{a}-\hat{b}|=\sqrt{1+1-2\cos \left(\frac{2\pi }{3}\right)}=\sqrt{3}$
$\therefore \left(S1\right)$ is correct.

Now, $\left(S2\right):$
Projection of $\hat{a}$ on $(\hat{a}+\hat{b})$
$=\left|\frac{\hat{a}\cdot (\hat{a}+\hat{b})}{|\hat{a}+\hat{b}|}\right|=\frac{1}{2}$.

$\therefore \left(S2\right)$ is also correct.