The angle between the lines x=1, y=2 and y=-1, z=0 is

Options

A. `30^(@)`
B. `60^(@)`
C. `90^(@)`
D. `0^(@)`

Explanation

To find the angle between the given lines \( x = 1, y = 2 \) and \( y = -1, z = 0 \), we can follow these steps:

### Step 1: Write the equations of the lines in parametric form
The first line can be represented in parametric form as:
\[ \frac{x - 1}{0} = \frac{y - 2}{0} = \frac{z - 0}{1} \]
This indicates that the direction ratios for the first line are \( (0, 0, 1) \).

The second line can be represented as:
\[ \frac{x - 0}{1} = \frac{y + 1}{0} = \frac{z - 0}{0} \]
This indicates that the direction ratios for the second line are \( (1, 0, 0) \).

### Step 2: Identify the direction ratios
From the parametric forms, we have:
- For Line 1: Direction ratios \( (L_1, M_1, N_1) = (0, 0, 1) \)
- For Line 2: Direction ratios \( (L_2, M_2, N_2) = (1, 0, 0) \)

### Step 3: Use the formula for the cosine of the angle between two lines
The cosine of the angle \( \theta \) between two lines can be found using the formula:
\[ \cos \theta = \frac{L_1 L_2 + M_1 M_2 + N_1 N_2}{\sqrt{L_1^2 + M_1^2 + N_1^2} \cdot \sqrt{L_2^2 + M_2^2 + N_2^2}} \]

### Step 4: Calculate the dot product
Calculating the dot product:
\[ L_1 L_2 + M_1 M_2 + N_1 N_2 = 0 \cdot 1 + 0 \cdot 0 + 1 \cdot 0 = 0 \]

### Step 5: Calculate the magnitudes of the direction ratios
Calculating the magnitudes:
\[ \sqrt{L_1^2 + M_1^2 + N_1^2} = \sqrt{0^2 + 0^2 + 1^2} = \sqrt{1} = 1 \]
\[ \sqrt{L_2^2 + M_2^2 + N_2^2} = \sqrt{1^2 + 0^2 + 0^2} = \sqrt{1} = 1 \]

### Step 6: Substitute values into the cosine formula
Substituting the values into the cosine formula:
\[ \cos \theta = \frac{0}{1 \cdot 1} = 0 \]

### Step 7: Find the angle \( \theta \)
Since \( \cos \theta = 0 \), it implies:
\[ \theta = 90^\circ \]

### Conclusion
The angle between the lines \( x = 1, y = 2 \) and \( y = -1, z = 0 \) is \( 90^\circ \).

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