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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
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B
π2
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C
π3
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D
π4
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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 vectors
So, $$\cos\theta= \dfrac{1}{2}$$
$$\therefore \theta=\dfrac{\pi}{3}$$
Hence, option '$$C'$$ is correct. 

Mathematics

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