Question

Let $\hat a$ and $\hat b$ be two unit vectors such that $| (a+b)+2(a × b)|$ =$2$. If $θ ∈ (0, π)$ is the angle between a and b, then among the statements: $(S1)$: $2|a×b|=|a-b|$ $(S2)$: The projection of a on $(\hat a+\hat b)$ is $\frac{1}{2}$

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

Answer 1

Answer: (C)

Both (S1) and (S2) are true

Answer 2

$\because |(\hat a+\hat b)+2(\hat a × \hat b)| = 2, θ ∈ (0, π)$

$⇒ |\hat a+\hat b+2(\hat a×\hat b)|2=4$

$⇒ |\hat a|^2+|\hat b|^2+4|\hat a×\hat b|^2+2a.b=4$

$∴ cosθ = cos2θ$

$∴ θ = \frac{2π}{3}$

where $θ$ is angle between $a$ and $b$.

$∴ 2|\hat a× \hat b|=\sqrt3=|\hat a-\hat b|$

(S1) is proved to be true.

As well as the projection of $\hat a$ on $(\hat a+\hat b) = |\frac{\hat a.(\hat a+\hat b)}{|\hat a+\hat b|}|=\frac{1}{2}$

(S2) is also proved to be true.

Hence, the correct option is (C): Both (S1) and (S2) are true