Question

If a line makes angle \[\frac{\pi}{3} \text{ and } \frac{\pi}{4}\]  with x-axis and y-axis respectively, then the angle made by the line with z-axis is

Options

• π/2

• π/3

•  π/4 

•  5π/12

Answer 1

 π/3 

If a line makes angles α, β and γ with the axes, then \[\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\]

Here,

\[\alpha = \frac{\pi}{3}\]

\[\beta = \frac{\pi}{4}\]

Now, 

\[\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\]

\[ \Rightarrow \cos^2 \frac{\pi}{3} + \cos^2 \frac{\pi}{4} + \cos^2 \gamma = 1\]

\[ \Rightarrow \frac{1}{4} + \frac{1}{2} + \cos^2 \gamma = 1\]

\[ \Rightarrow \cos^2 \gamma = 1 - \frac{3}{4}\]

\[ \Rightarrow \cos^2 \gamma = \frac{1}{4}\]

\[ \Rightarrow \cos \gamma = \frac{1}{2}\]

\[ \Rightarrow \gamma = \frac{\pi}{3}\]