Three Dimensional Geometry Question 395
Question: Consider the following relations among the angles $ \alpha $ , $ \beta $ and $ \gamma $ made by a vector with the coordinate axes
I. $ \cos 2\alpha +\cos 2\beta +\cos 2\gamma =-1 $ II. $ {{\sin }^{2}}\alpha +{{\sin }^{2}}\beta +{{\sin }^{2}}\gamma =1 $ Which of the above is/are correct?
Options:
A) Only I
B) Only II
C) Both I and II
D) Neither I nor II
Show Answer
Answer:
Correct Answer: A
Solution:
[a] We have. $ {{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\cos }^{2}}\gamma =1 $ ??(i)
$ \Rightarrow 2{{\cos }^{2}}\alpha +2{{\cos }^{2}}\beta +2{{\cos }^{2}}\gamma =2 $
$ \Rightarrow 2{{\cos }^{2}}\alpha -1+2{{\cos }^{2}}\beta -1+2{{\cos }^{2}}\gamma -1=2-3 $
$ \Rightarrow \cos 2\alpha +\cos 2\beta +\cos 2\gamma =-1 $ Hence statement- I is correct. And now from (i), $ 1-{{\sin }^{2}}\alpha +1-{{\sin }^{2}}\beta +1-{{\sin }^{2}}\gamma =1 $
$ \Rightarrow {{\sin }^{2}}\alpha +{{\sin }^{2}}\beta +{{\sin }^{2}}\gamma =2 $ Hence, only statement I is correct.
BETA
