Question

The angle between the lines \( x = y, z = 0 \) and \( y = 0, z = 0 \) is

• A. \( 30^\circ \)
• B. \( 45^\circ \)
• C. \( 60^\circ \)
• D. \( 90^\circ \)

Hint:

The $x$-axis has direction cosines $(1, 0, 0)$, $y$-axis $(0, 1, 0)$, and $z$-axis $(0, 0, 1)$.

Answer 1

The Correct Option is B

Step 1: Find DRs of Line 1
$x = y, z = 0$ is a line in the $xy$-plane. DRs: $(1, 1, 0)$.
Step 2: Find DRs of Line 2
$y = 0, z = 0$ is the $x$-axis. DRs: $(1, 0, 0)$.
Step 3: Apply Cosine Formula
$\cos \theta = \frac{|(1)(1) + (1)(0) + (0)(0)|}{\sqrt{1^2+1^2+0^2}\sqrt{1^2+0^2+0^2}} = \frac{1}{\sqrt{2} \cdot 1} = \frac{1}{\sqrt{2}}$.
Step 4: Conclusion
$\cos \theta = 1/\sqrt{2} \implies \theta = 45^\circ$.
Final Answer:(B)