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