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 the above

Explanation

To find the angle between the two lines given by the equations:

1. \(\frac{x-2}{1} = \frac{y+1}{-2} = \frac{z+2}{1}\)
2. \(\frac{x-1}{1} = \frac{2y+3}{3} = \frac{z+5}{2}\)

we will follow these steps:

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

From the first line, we can extract the direction ratios:
- \(a_1 = 1\)
- \(b_1 = -2\)
- \(c_1 = 1\)

From the second line, we need to manipulate it to find the direction ratios. The second line can be rewritten as:
\[ \frac{x-1}{1} = \frac{y + \frac{3}{2}}{\frac{3}{2}} = \frac{z+5}{2} \]
This means we can express \(y\) in terms of \(x\) and \(z\) as follows:
- \(x - 1 = t\) (where \(t\) is a parameter)
- \(y + \frac{3}{2} = \frac{3}{2}t\) or \(y = \frac{3}{2}t - \frac{3}{2}\)
- \(z + 5 = 2t\) or \(z = 2t - 5\)

This gives us the direction ratios for the second line:
- \(a_2 = 1\)
- \(b_2 = \frac{3}{2}\)
- \(c_2 = 2\)

### Step 2: Use the formula for the angle between two lines

The formula for the cosine of the angle \(\theta\) between two lines with direction ratios \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\) is given by:

\[ \cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \cdot \sqrt{a_2^2 + b_2^2 + c_2^2}} \]

### Step 3: Substitute the values into the formula

Substituting the values we found:
- \(a_1 = 1, b_1 = -2, c_1 = 1\)
- \(a_2 = 1, b_2 = \frac{3}{2}, c_2 = 2\)

Calculating the numerator:
\[ a_1 a_2 + b_1 b_2 + c_1 c_2 = 1 \cdot 1 + (-2) \cdot \frac{3}{2} + 1 \cdot 2 = 1 - 3 + 2 = 0 \]

Calculating the denominator:
\[ \sqrt{a_1^2 + b_1^2 + c_1^2} = \sqrt{1^2 + (-2)^2 + 1^2} = \sqrt{1 + 4 + 1} = \sqrt{6} \]
\[ \sqrt{a_2^2 + b_2^2 + c_2^2} = \sqrt{1^2 + \left(\frac{3}{2}\right)^2 + 2^2} = \sqrt{1 + \frac{9}{4} + 4} = \sqrt{\frac{4 + 9 + 16}{4}} = \sqrt{\frac{29}{4}} = \frac{\sqrt{29}}{2} \]

### Step 4: Calculate \(\cos \theta\)

Now substituting back into the formula:
\[ \cos \theta = \frac{0}{\sqrt{6} \cdot \frac{\sqrt{29}}{2}} = 0 \]

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

Since \(\cos \theta = 0\), this implies:
\[ \theta = \frac{\pi}{2} \text{ radians} \quad \text{(90 degrees)} \]

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

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