Physics Two Dimensional Motion Question 138
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} $ 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} ).( 5\hat{a}-4,\hat{b} )=0 $ $ \Rightarrow 5+\text{6},\hat{a}\cdot\hat{b}-8=0\text{}( \
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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