Question

If for unit vectors $\overline{a}$ and $\overline{b}$, $\overline{a}+2\overline{b}$ and $5\overline{a}-4\overline{b}$ are each-other, then $(\overline{a}\wedge \overline{b})=$ _____________.

• A. ${\cos }^{-1}\frac{1}{3}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{4}$
• D. ${\cos }^{-1}\frac{2}{7}$

Answer 1

The correct option is B $\frac{\pi }{3}$

$\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vector , and $5\overrightarrow{a}-4\overrightarrow{b}$, $\overrightarrow{a}+2\overrightarrow{b}$ are perpendicular so

$\Rightarrow$ $(5\overrightarrow{a}-4\overrightarrow{b}).(\overrightarrow{a}+\overrightarrow{2b})=0$

$\Rightarrow$ $5a^2+10ab-4ab-8b^2$ = 0

$\Rightarrow$ $5a^2+6ab-8b^2$ = 0....equation 1

now we know $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vector so

$(\overrightarrow{a}.\overrightarrow{b})=\frac{1}{2}$

so angle between $\overrightarrow{a}$ , $\overrightarrow{b}$ is $\frac{\pi }{3}$