Find the angle between the following pair of lines ,
(i) `(x - 2)/(2) = (y + 3)/(5) = (z + 3)/(-3) and (x + 2)/(-1) = (y - 4)/(8) = (z - 5 )/(4)`
(ii) `(x)/(2) = (y)/(2) = (z)/(1) and (x -5)/(4) = (y - 2)/(1) = (z -3)/(8)` .

Answer

To find the angle between the given pairs of lines, we will follow these steps for each part of the question.

### Part (i)
The equations of the lines are given as:
1. \((x - 2)/(2) = (y + 3)/(5) = (z + 3)/(-3)\)
2. \((x + 2)/(-1) = (y - 4)/(8) = (z - 5)/(4)\)

**Step 1: Identify Direction Ratios**
From the first line, we can extract the direction ratios:
- \(d_1 = (2, 5, -3)\)

From the second line, we can extract the direction ratios:
- \(d_2 = (-1, 8, 4)\)

**Step 2: Calculate the Dot Product**
The dot product \(d_1 \cdot d_2\) is calculated as follows:
\[ d_1 \cdot d_2 = (2)(-1) + (5)(8) + (-3)(4) \]
\[ = -2 + 40 - 12 = 26 \]

**Step 3: Calculate the Magnitudes**
Now, we calculate the magnitudes of \(d_1\) and \(d_2\):
\[ |d_1| = \sqrt{2^2 + 5^2 + (-3)^2} = \sqrt{4 + 25 + 9} = \sqrt{38} \]
\[ |d_2| = \sqrt{(-1)^2 + 8^2 + 4^2} = \sqrt{1 + 64 + 16} = \sqrt{81} = 9 \]

**Step 4: Use the Cosine Formula**
Using the formula for the cosine of the angle \(\theta\) between the two lines:
\[ \cos \theta = \frac{d_1 \cdot d_2}{|d_1| |d_2|} = \frac{26}{\sqrt{38} \cdot 9} \]

**Step 5: Calculate \(\theta\)**
To find \(\theta\), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{26}{9\sqrt{38}}\right) \]

### Part (ii)
The equations of the lines are given as:
1. \((x)/(2) = (y)/(2) = (z)/(1)\)
2. \((x - 5)/(4) = (y - 2)/(1) = (z - 3)/(8)\)

**Step 1: Identify Direction Ratios**
From the first line, we extract the direction ratios:
- \(d_1 = (2, 2, 1)\)

From the second line, we extract the direction ratios:
- \(d_2 = (4, 1, 8)\)

**Step 2: Calculate the Dot Product**
The dot product \(d_1 \cdot d_2\) is calculated as follows:
\[ d_1 \cdot d_2 = (2)(4) + (2)(1) + (1)(8) \]
\[ = 8 + 2 + 8 = 18 \]

**Step 3: Calculate the Magnitudes**
Now, we calculate the magnitudes of \(d_1\) and \(d_2\):
\[ |d_1| = \sqrt{2^2 + 2^2 + 1^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]
\[ |d_2| = \sqrt{4^2 + 1^2 + 8^2} = \sqrt{16 + 1 + 64} = \sqrt{81} = 9 \]

**Step 4: Use the Cosine Formula**
Using the formula for the cosine of the angle \(\theta\) between the two lines:
\[ \cos \theta = \frac{d_1 \cdot d_2}{|d_1| |d_2|} = \frac{18}{3 \cdot 9} = \frac{18}{27} = \frac{2}{3} \]

**Step 5: Calculate \(\theta\)**
To find \(\theta\), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{2}{3}\right) \]