Kinematics Question 590
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} $ is:
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} \cdot \vec{d}=0\Rightarrow ( \hat{a}+2\hat{b} )\cdot( 5\hat{a}-4\hat{b} )=0 $ $ \Rightarrow 5+6\hat{a}\cdot\hat{b}-8=0$
$( \therefore \vec{a} \cdot \vec{a} = 1 ) $ $ \Rightarrow \hat{a} \cdot \hat{b} = \frac{1}{2} \Rightarrow \theta = \frac{\pi }{3} $
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