Question

If a line makes an angle of $\frac{\pi}{4}$ with the positive directions of both $x$-axis and $z$-axis, then the angle which it makes with the positive direction of $y$-axis is:

• A. $0$
• B. $\frac{\pi}{4}$
• C. $\frac{\pi}{2}$
• D. $\pi$

Hint:

The direction cosines of a line satisfy $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$. Use known angles to compute the unknown.

Answer 1

The Correct Option is C

The angles $\alpha, \beta, \gamma$ made by the line with the $x$-axis, $y$-axis, and $z$-axis respectively, satisfy the equation for direction cosines: $$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1.$$ Given that $\alpha = \frac{\pi}{4}$ and $\gamma = \frac{\pi}{4}$, we calculate: $$\cos \alpha = \cos \gamma = \frac{\sqrt{2}}{2}.$$ Substitute these values into the equation: $$\left(\frac{\sqrt{2}}{2}\right)^2 + \cos^2 \beta + \left(\frac{\sqrt{2}}{2}\right)^2 = 1.$$ Simplify: $$\frac{1}{2} + \cos^2 \beta + \frac{1}{2} = 1 \quad \Rightarrow \quad \cos^2 \beta = 0.$$ This implies: $$\cos \beta = 0 \quad \Rightarrow \quad \beta = \frac{\pi}{2}.$$
Final Answer: $\boxed{\frac{\pi}{2}}$.