A line makes an angle of π4with the positive directions of each of y-and z-axes. Then the angle that the line makes with the positive direction of x-axis, is

# A line makes an angle of $\frac{\pi }{4}$with the positive directions of each of $y-$and $z-$axes. Then the angle that the line makes with the positive direction of $x-$axis, is

1. A

$\pi$

2. B

$\frac{\pi }{4}$

3. C

$\frac{\pi }{6}$

4. D

$\frac{\pi }{2}$

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### Solution:

If $\alpha ,\beta ,\gamma$ are the angles made by a line ‘L’ with the coordinate axes in the positive direction then ${\mathrm{cos}}^{2}\alpha +{\mathrm{cos}}^{2}\beta +{\mathrm{cos}}^{2}\gamma =1$

Given $\beta =\frac{\pi }{4},\gamma =\frac{\pi }{4}$

Hence, ${\mathrm{cos}}^{2}\alpha +\frac{1}{2}+\frac{1}{2}=1⇒\mathrm{cos}\gamma =0$

Therefore, $\gamma =\overline{)\frac{\pi }{2}}$

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