Question

If $a$ and $b$ are two unit vectors inclined at an angle $\frac{\pi }{3}$, then $\{a\times (b+a\times b)\}\cdot b$ is equal to

• A. $\frac{1}{4}$
• B. $\frac{-3}{4}$
• C. $\frac{3}{4}$
• D. $\frac{1}{2}$

Answer 1

Answer: (B)

$\frac{-3}{4}$

Given, $\left|a\right|=\left|b\right|=1$ and $a\cdot b=\cos \frac{\pi }{3}$
Consider,
$\{a\times (b+a\times b)\}\cdot b=\{a\times b+a\times (a\times b)\}\cdot b$
$=(a\times b)\cdot b+\{(a\cdot b)\cdot a-(a\cdot a)\cdot b\}\cdot b$
$=\left[\begin{array}{ccc}a& b& b\end{array}\right]+{(a\cdot b)}^2-{\left|a\right|}^2{\left|b\right|}^2$
$=0+{\cos }^2\frac{\pi }{3}-1=\frac{1}{4}-1=\frac{-3}{4}$