What is the angle between the lines `(x-2)/1=(y+1)/(-2)=(z+2)/1 and (x-1)/1=(2y+3)/3=(z+5)/2`

Options

A. `pi/2`
B. `pi/3`
C. `pi/6`
D. None of these

Answer

To find the angle between the given lines, we will follow these steps:

### Step 1: Identify the Direction Ratios of the Lines
The equations of the lines are given in symmetric form:
1. Line 1: \((x-2)/1 = (y+1)/(-2) = (z+2)/1\)
2. Line 2: \((x-1)/1 = (2y+3)/3 = (z+5)/2\)

From these equations, we can extract the direction ratios (or direction vectors) for each line.

- For Line 1, the direction ratios are \(1, -2, 1\).
- For Line 2, we can rewrite the second part to find the direction ratios. The direction ratios are \(1, 3/2, 2\).

### Step 2: Write the Direction Vectors
Now we can express the direction vectors:
- Let \( \mathbf{a} = \langle 1, -2, 1 \rangle \) for Line 1.
- Let \( \mathbf{b} = \langle 1, \frac{3}{2}, 2 \rangle \) for Line 2.

### Step 3: Use the Formula for the Angle Between Two Vectors
The angle \( \theta \) between two vectors can be found using the formula:
\[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} \]
where \( \mathbf{a} \cdot \mathbf{b} \) is the dot product of the vectors and \( |\mathbf{a}| \) and \( |\mathbf{b}| \) are the magnitudes of the vectors.

### Step 4: Calculate the Dot Product
The dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as follows:
\[ \mathbf{a} \cdot \mathbf{b} = (1)(1) + (-2)(\frac{3}{2}) + (1)(2) = 1 - 3 + 2 = 0 \]

### Step 5: Calculate the Magnitudes of the Vectors
Now we calculate the magnitudes:
\[ |\mathbf{a}| = \sqrt{1^2 + (-2)^2 + 1^2} = \sqrt{1 + 4 + 1} = \sqrt{6} \]
\[ |\mathbf{b}| = \sqrt{1^2 + \left(\frac{3}{2}\right)^2 + 2^2} = \sqrt{1 + \frac{9}{4} + 4} = \sqrt{1 + 2.25 + 4} = \sqrt{7.25} = \frac{\sqrt{29}}{2} \]

### Step 6: Substitute into the Formula
Now substitute the values into the formula:
\[ \cos \theta = \frac{0}{\sqrt{6} \cdot \frac{\sqrt{29}}{2}} = 0 \]

### Step 7: Find the Angle
Since \( \cos \theta = 0 \), we have:
\[ \theta = \cos^{-1}(0) = \frac{\pi}{2} \]

### Final Answer
The angle between the lines is \( \frac{\pi}{2} \) radians or 90 degrees.

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