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 $