First slide

Direction ratios and direction cosines

Question

A line makes an angle of $\frac{π}{3}$ with the positive direction of $x -$ axis and an angle of $\frac{π}{4}$ with the positive direction of $z -$ axis. Then the angle that the line makes with the positive direction of $y -$ axis, is

Very Easy

A

$\frac{π}{4}$
B

$\frac{π}{6}$
C

$\frac{π}{3}$
D

$\frac{π}{2}$

Solution

If $α, β, γ$ are the angles made by a line ‘L’ with the coordinate axes in the positive direction then $\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$
Given $α = \frac{π}{3}, γ = \frac{π}{4}$
Hence, $\frac{1}{4} + \cos^{2} β + \frac{1}{2} = 1 \Rightarrow \cos β = \frac{1}{\sqrt{2}}$
Therefore, $β = \frac{π}{4}$

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