First slide

Direction ratios and direction cosines

Question

If a line passing through origin is inclined at $60 °$ and $45 °$ to $X -$ and $Y -$ axes respectively, Then the angle at which it is inclined to $Z -$ axis, is

Easy

A

$30 °$
B

$60 °$
C

$45 °$
D

$90 °$

Solution

If $α, β, γ$ are the angles made by a line ‘L’ with the coordinate axes in the positive direction then $\cos^{2} α + \cos^{2} β + \cos^{2} γ = 1$
Here, $α = 60 °, β = 45 °$
Hence,
$\begin{matrix} \cos^{2} (60 °) + \cos^{2} (45 °) + \cos^{2} γ = 1 \\ \frac{1}{4} + \frac{1}{2} + \cos^{2} γ = 1 \\ \cos^{2} γ = \frac{1}{4} \\ \cos γ = \pm \frac{1}{2} \end{matrix}$
Therefore, $γ = 60 ° or 120 °$

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