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

Options

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

Answer

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

### Step 1: Identify the Direction Ratios of Each Line
1. **Line 1**: The equations \( x = 1 \) and \( y = 2 \) imply that the line is parallel to the z-axis. Therefore, the direction ratios can be taken as:
\[ \text{Direction Ratios of Line 1} = (0, 0, 1) \]

2. **Line 2**: The equations \( y = -1 \) and \( z = 0 \) imply that the line is parallel to the x-axis. Therefore, the direction ratios can be taken as:
\[ \text{Direction Ratios of Line 2} = (1, 0, 0) \]

### Step 2: Use the Dot Product to Find the Angle
The angle \( \theta \) between two lines can be found using the formula:
\[ \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} \]
where \( (a_1, b_1, c_1) \) and \( (a_2, b_2, c_2) \) are the direction ratios of the two lines.

Substituting the direction ratios:
- For Line 1: \( (0, 0, 1) \) → \( a_1 = 0, b_1 = 0, c_1 = 1 \)
- For Line 2: \( (1, 0, 0) \) → \( a_2 = 1, b_2 = 0, c_2 = 0 \)

### Step 3: Calculate the Dot Product
Calculate the dot product:
\[ a_1 a_2 + b_1 b_2 + c_1 c_2 = 0 \cdot 1 + 0 \cdot 0 + 1 \cdot 0 = 0 \]

### Step 4: Calculate the Magnitudes
Calculate the magnitudes:
\[ \sqrt{a_1^2 + b_1^2 + c_1^2} = \sqrt{0^2 + 0^2 + 1^2} = \sqrt{1} = 1 \]
\[ \sqrt{a_2^2 + b_2^2 + c_2^2} = \sqrt{1^2 + 0^2 + 0^2} = \sqrt{1} = 1 \]

### Step 5: Substitute into the Cosine Formula
Substituting into the cosine formula:
\[ \cos \theta = \frac{0}{1 \cdot 1} = 0 \]

### Step 6: Find the Angle
Since \( \cos \theta = 0 \), we have:
\[ \theta = 90^\circ \]

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

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