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Mathematics
If a line makes an angle of π/3 with each of x and and y-axis, then the acute angle made by z-axis is
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Q. If a line makes an angle of $\pi/3$ with each of $x$ and and $y$-axis, then the acute angle made by $z$-axis is
KCET
KCET 2020
Three Dimensional Geometry
A
$\frac{\pi}{4}$
29%
B
$\frac{\pi}{6}$
27%
C
$\frac{\pi}{3}$
31%
D
$\frac{\pi}{2}$
14%
Solution:
Given $\alpha = \beta = \pi/3$
Let acute angle made by $z$ -axis be $y$
Then, $cos^2 \alpha + cos^2 \beta + cos^2 \gamma$
$\Rightarrow (\frac{1}{2})^2+(\frac{1}{2})^2 + cos^2 \gamma = 1$
$\Rightarrow \frac{1}{4}+\frac{1}{4} + cos^2 \gamma = 1 $
$\Rightarrow cos^2 \gamma = \frac{1}{2}$
$\Rightarrow cos \gamma = \pm \frac{1}{\sqrt 2}$
$\Rightarrow \gamma = \frac{\pi}{4}$
$[\therefore y $ is acute]