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 :
$$
(\mathrm{S} 1): 2|\hat{\mathrm{a}} \times \hat{\mathrm{b}}|=|\hat{\mathrm{a}}-\hat{\mathrm{b}}|
$$
$(\mathrm{S} 2)$ : The projection of $\hat{a}$ on $(\hat{a}+\hat{b})$ is $\frac{1}{2}$
Select the correct option:
A
Only (S1) is true
B
Only (S2) is true
C
Both (S1) and (S2) are true
D
Both (S1) and (S2) are false
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
$$
\begin{aligned}
& \mid a+b)+2(a \times b) \mid=2, \theta e(0, \pi) \\
& ((a+b)+2(a \times b))+(a+b)+2(a \times b))=4 \\
& |a+b|+|a \times b|+0=4
\end{aligned}
$$
Let the angle be $\theta$ between a and 6
$$
\begin{aligned}
& 2+2000 \theta+4 \sin ^2 \theta=4 \\
& 2+2000 \theta-400 a^2 \theta=0
\end{aligned}
$$
Let cose = t then
$$
\begin{aligned}
& 2 t^2-t-1=0 \\
& 2 t^2-2 t+t-1=0 \\
& 2 t(t-1)+(t-1)=0 \\
& (2 t+1)(t-1)=0 \\
& t=-\frac{1}{2} \quad \text { or } \quad t=1
\end{aligned}
$$
$\cos \theta=-\frac{1}{2}$
[not possiblic as $\theta$ e ( 0 , xil
$$
\theta=\frac{2 \pi}{3}
$$
Now,
Si $2|\sin |=2 \sin \left(\frac{2 \pi}{3}\right)$
$$
\begin{aligned}
& \text { 1-6 } \sqrt{1+1-2 \cos \left(\frac{2 \pi}{3}\right)} \\
& =\sqrt{2-2 \cdot\left(-\frac{1}{2}\right)} \\
& =\sqrt{5}
\end{aligned}
$$
Si is correct.
Se projection of a on $(\vec{a}+\vec{b})$.
$$
\begin{aligned}
& \frac{a(a+b)}{1+b}=\frac{1+\cos \left(\frac{2 \pi}{3}\right)}{\sqrt{2+2 \cos \frac{2 \pi}{3}}} \\
& =\frac{1-\frac{1}{2}}{\sqrt{1}} \\
& =\frac{1}{2}
\end{aligned}
$$
C Opton is true.
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