Vector Algebra Question 45
Question: If $ \hat{a}, $ $ \hat{b} $ and $ \hat{c} $ are three unit vectors, such that $ \hat{a}+\hat{b}+\hat{c} $ is also a unit vector and $ {\theta_1} $ , $ {\theta_2} $ and $ {\theta_3} $ are angles between the vectors $ \hat{a} $ , $ \hat{b} $ ; $ \hat{b} $ , $ \hat{c} $ and $ \hat{c} $ , $ \hat{a} $ , respectively, then among $ \theta _1^{{}} $ , $ \theta _2^{{}} $ and $ \theta _3^{{}} $
Options:
A) all are acute angles
B) all are right angles
C) at least one is obtuse angle
D) none of these
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Answer:
Correct Answer: C
Solution:
- [c] $ {{| \vec{a}+\vec{b}+\vec{c} |}^{2}}=1 $
$ \Rightarrow {{| {\vec{a}} |}^{2}}+{{| {\vec{b}} |}^{2}}+{{| {\vec{c}} |}^{2}}+2| {\vec{a}} || {\vec{b}} |\cos {\theta_1} $ $ +2| {\vec{b}} || {\vec{c}} |\cos {\theta_2}+2| {\vec{c}} || {\vec{a}} |\cos {\theta_3}=1 $
$ \Rightarrow \cos {\theta_1}+\cos {\theta_2}+\cos {\theta_3}=-1 $ Hence, one of $ {\theta_1},{\theta_2} $ and $ {\theta_3} $ should be an obtuse angle.
BETA
