Kinematics Question 590

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

Options:

A) $ \frac{\pi }{6} $

B)$ \frac{\pi }{2} $

C) $ \frac{\pi }{3} $

D)$ \frac{\pi }{4} $

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Answer:

Correct Answer: C

Solution:

[c] Let $ \vec{c}=\hat{a}+2\hat{b} $ and $ \vec{d}=5\hat{a}-4\hat{b} $

Since $ \vec{c} $ and $ \vec{d} $ are perpendicular to each other

$ \therefore \vec{c}{.}{\vec{d}=0}\Rightarrow ( {\hat{a}+2}\hat{b} ).( {5\hat{a}}-4\hat{b} )=0 $ $ \Rightarrow 5+{6}\hat{a}{.\hat{b}}-8=0{}$

$( \therefore \vec{a}{.}\vec{a}{=1} ) $ $ \Rightarrow \hat{a}{.\hat{b}=}\frac{1}{2}\Rightarrow \theta =\frac{\pi }{3} $



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