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} $
Show Answer
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} $
