Three Dimensional Geometry Question 13
Question: A line makes $ 45{}^\circ $ with positive x-axis and makes equal angles with positive y, z axes, respectively. What is the sum of the three angles which the line makes with positive x, y and z axes?
Options:
A) $ 180{}^\circ $
B) $ 165{}^\circ $
C) $ 150{}^\circ $
D) $ 135{}^\circ $
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Answer:
Correct Answer: B
Solution:
[b] We know that sum of square of direction cosines = 1 i.e. $ {{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\cos }^{2}}\gamma =1 $
$ \Rightarrow {{\cos }^{2}}45{}^\circ +{{\cos }^{2}}\beta +{{\cos }^{2}}\beta =1 $ (As given $ \alpha =45{}^\circ and\beta =\gamma $ )
$ \Rightarrow \frac{1}{2}+2{{\cos }^{2}}\beta =1 $
$ \Rightarrow {{\cos }^{2}}\beta =\frac{1}{4} $
$ \Rightarrow \cos \beta =\pm \frac{1}{2}, $ Negative value is discarded, since the line makes angle with positive axes. Hence, $ \cos \beta =\frac{1}{2} $
$ \Rightarrow \cos \beta =\cos 60{}^\circ $ $ \beta =60{}^\circ $
$ \therefore $ Required sum $ =\alpha +\beta +\gamma =45{}^\circ +60{}^\circ +60{}^\circ $ $ =165{}^\circ $
BETA
