Question

If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis.

Answer 1

Let the direction cosines of the line be l, m and n.
We know that l² + m² + n² = 1.
Let the line make angle θ with the positive direction of the z-axis.

$$\begin{gathered} \alpha =90°,\beta =60°,\gamma =\theta \\ \mathrm{So},{\cos }^{2}90°+{\cos }^{2}60°+{\cos }^2\theta =1 \\ \Rightarrow 0+{\left(\frac{1}{2}\right)}^2+{\cos }^2\theta =1 \\ \Rightarrow {\cos }^2\theta =1-\frac{1}{4} \\ \Rightarrow {\cos }^2\theta =\frac{3}{4} \\ \Rightarrow \cos \theta =\pm \frac{\sqrt{3}}{2} \\ \Rightarrow \theta =30°\mathrm{or}150° \end{gathered}$$


cos2α+cos2β+cos2γ=1Here α=60 and β=45 and γ= θSo cos260+cos245+cos2θ=1cos2θ=1−14−12=14cosθ=±12So θ= 60 degree or 120.and here it is given that we have to find the angle made by negative z −axisSo cosθ=−12θ=120 degree