Question

Let $\overrightarrow{a} $ and $\overrightarrow{b} $ be two unit vectors. If the vectors $\overrightarrow{c} =\overrightarrow{a}+2\,\overrightarrow{b} $ and $\overrightarrow{d} =5\,\overrightarrow{a}-4\,\overrightarrow{b} $ are perpendicular to each other, then the angle between $\overrightarrow{a} $ and $\overrightarrow{b} $ is .

• A. $\frac{\pi}{2}$
• B. $\frac{\pi}{3}$
• C. $\frac{\pi}{4}$
• D. $\frac{\pi}{6}$

Answer 1

Answer: (B)

Since $\vec{c}$, $\vec{d}$ are perpendicular
$\therefore \vec{c}\cdot\vec{d}=0$
$\therefore \left(\vec{a}+2\,\vec{b}\right).\left(5\,\vec{a}-4\,\vec{b}\right)=0$
$\Rightarrow 5\,\vec{a}^{2}-4\,\vec{a} \cdot \vec{b}+10\,\vec{b} \cdot \vec{a}-8\,\vec{b}^{2}=0$
$\Rightarrow 5\left(1\right)+6\,\vec{a}\cdot\vec{b}-8\left(1\right)=0$
$\Rightarrow 6\,\vec{a}\cdot\vec{b}=8-5=3$
$\Rightarrow \vec{a}\cdot\vec{b}=\frac{3}{6}=\frac{1}{2}$
$\Rightarrow \left(1\right)\,\left(1\right)\,cos\,\theta=\frac{1}{2}$
$\Rightarrow \theta=\frac{\pi}{3}$