Question

$a,b$ and $c$ are three unit vectors such that no two of them are collinear. If $b=2\{a\times (b\times c)\}$ and $\alpha$ is the angle between $a,c$ and $\beta$ is the angle between $a,b$, then $\cos (\alpha +\beta )=$

• A. $\frac{\sqrt{3}}{2}$
• B. $-\frac{\sqrt{3}}{2}$
• C. $\frac{1}{2}$
• D. $-\frac{1}{2}$

Answer 1

Answer: (B)

We have,
$b=2\{a\times (b\times c)\}$
$b=2\{(a\cdot c)b-(a\cdot b)c\}$
$b=2(a\cdot c)b-2(a\cdot b)c$
On comparing $b$ and $c$, we get
$\therefore 2(a\cdot c)=1$ and $a\cdot b=0$
$\alpha$ is a angle between $a,c$ and $\beta$ is the angle between $a,b$. $|a||c|\cos \alpha =\frac{1}{2},a\cdot b=0$
$\cos \alpha =\cos \frac{\pi }{3},\text{ and }\cos \beta =\cos \frac{\pi }{2}$
$\therefore \alpha =\pi /3$ and $\beta =\frac{\pi }{2}$
$\Rightarrow \cos (\alpha +\beta )=-\sin \pi /3=-\sqrt{3}/2$