Question

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

A
π6
B
π2
C
π3
D
π4

Solution

## The correct option is D $$\displaystyle \dfrac{\pi}{3}$$Given that $$\vec c$$ and $$\vec d$$ are Perpendicular$$\therefore$$ $$\vec{c}.\vec{d}=0$$Putting values$$\Rightarrow 5|\vec{a}|^2 +6ab-8|b|^2=0$$$$\Rightarrow 6\vec{a}.\vec{b}=3$$$$\Rightarrow \displaystyle \vec{a}.\vec{b}=\dfrac{1}{2}$$$$=|\vec a||\vec b|\cos\theta=\dfrac{1}{2}$$Since $$\vec a$$ and $$\vec b$$ are Unit vectorsSo, $$\cos\theta= \dfrac{1}{2}$$$$\therefore \theta=\dfrac{\pi}{3}$$Hence, option '$$C'$$ is correct. Mathematics

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