Find the acute angle between the lines x=y: z=0 and x=0z=0

Answer

To find the acute angle between the lines given by the equations \( x = y: z = 0 \) and \( x = 0: z = 0 \), we can follow these steps:

### Step 1: Identify the direction ratios of the lines

1. For the first line \( L_1: x = y, z = 0 \):
- We can express this in parametric form as:
- \( x = t \)
- \( y = t \)
- \( z = 0 \)
- The direction ratios for \( L_1 \) are \( (1, 1, 0) \).

2. For the second line \( L_2: x = 0, z = 0 \):
- We can express this in parametric form as:
- \( x = 0 \)
- \( y = s \)
- \( z = 0 \)
- The direction ratios for \( L_2 \) are \( (0, 1, 0) \).

### Step 2: Use the direction ratios to find the angle between the lines

To find the angle \( \theta \) between the two lines, we can use the formula for the cosine of the angle between two vectors:

\[ \cos \theta = \frac{L_1 \cdot L_2}{|L_1| |L_2|} \]

Where \( L_1 \cdot L_2 \) is the dot product of the direction ratios, and \( |L_1| \) and \( |L_2| \) are the magnitudes of the direction ratios.

### Step 3: Calculate the dot product and magnitudes

1. **Dot Product**:
\[ L_1 \cdot L_2 = (1)(0) + (1)(1) + (0)(0) = 0 + 1 + 0 = 1 \]

2. **Magnitude of \( L_1 \)**:
\[ |L_1| = \sqrt{1^2 + 1^2 + 0^2} = \sqrt{1 + 1 + 0} = \sqrt{2} \]

3. **Magnitude of \( L_2 \)**:
\[ |L_2| = \sqrt{0^2 + 1^2 + 0^2} = \sqrt{0 + 1 + 0} = 1 \]

### Step 4: Substitute into the cosine formula

Now substituting into the cosine formula:

\[ \cos \theta = \frac{1}{\sqrt{2} \cdot 1} = \frac{1}{\sqrt{2}} \]

### Step 5: Find the angle \( \theta \)

To find \( \theta \), we take the inverse cosine:

\[ \theta = \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) \]

This gives us:

\[ \theta = 45^\circ \]

### Conclusion

The acute angle between the lines \( x = y: z = 0 \) and \( x = 0: z = 0 \) is \( 45^\circ \).

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