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: Understanding the Question:
We need to find the direction ratios of each line and then use the angle formula.
Step 2: Detailed Explanation:
Line 1: \( x = y, z = 0 \). In symmetric form, \( \frac{x}{1} = \frac{y}{1} = \frac{z}{0} \).
DRs of line 1 are \( (1, 1, 0) \).
Line 2: \( y = 0, z = 0 \). This is the x-axis. In symmetric form, \( \frac{x}{1} = \frac{y}{0} = \frac{z}{0} \).
DRs of line 2 are \( (1, 0, 0) \).
Calculate the angle \( \theta \):
\[ \cos \theta = \frac{|(1)(1) + (1)(0) + (0)(0)|}{\sqrt{1^2+1^2+0^2}\sqrt{1^2+0^2+0^2}} = \frac{1}{\sqrt{2} \cdot 1} = \frac{1}{\sqrt{2}} \] Thus, \( \theta = 45^\circ \).
Step 4: Final Answer:
The angle is \( 45^\circ \).