Find the angle between the following pair of lines
`(x-2)/(2) = (y - 1)/(7) = (z + 3)/(-3) and (x -2)/(-1) = (2y-8)/(4) = (z+5)/(4)`
and check whether the lines are parallel or perpendicular .

Answer

To find the angle between the given pair of lines and check whether they are parallel or perpendicular, we will follow these steps:

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

The lines are given in symmetric form:

1. Line 1: \((x-2)/(2) = (y - 1)/(7) = (z + 3)/(-3)\)
2. Line 2: \((x -2)/(-1) = (2y-8)/(4) = (z+5)/(4)\)

From the first line, the direction ratios can be extracted directly from the denominators:
- For Line 1: The direction ratios are \(d_1 = (2, 7, -3)\).

For the second line, we need to simplify the equation:
- The second line can be rewritten as:
\[ \frac{x - 2}{-1} = \frac{y - 4}{2} = \frac{z + 5}{4} \]
Thus, the direction ratios are \(d_2 = (-1, 2, 4)\).

### Step 2: Write the direction ratios in vector form

Now we can express the direction ratios in vector form:
- \(D_1 = 2i + 7j - 3k\)
- \(D_2 = -i + 2j + 4k\)

### Step 3: Calculate the dot product of the direction vectors

The dot product \(D_1 \cdot D_2\) is calculated as follows:
\[ D_1 \cdot D_2 = (2)(-1) + (7)(2) + (-3)(4) \]
Calculating each term:
- \(2 \cdot -1 = -2\)
- \(7 \cdot 2 = 14\)
- \(-3 \cdot 4 = -12\)

Now, summing these results:
\[ D_1 \cdot D_2 = -2 + 14 - 12 = 0 \]

### Step 4: Calculate the magnitudes of the direction vectors

Next, we calculate the magnitudes of \(D_1\) and \(D_2\):
\[ |D_1| = \sqrt{2^2 + 7^2 + (-3)^2} = \sqrt{4 + 49 + 9} = \sqrt{62} \]
\[ |D_2| = \sqrt{(-1)^2 + 2^2 + 4^2} = \sqrt{1 + 4 + 16} = \sqrt{21} \]

### Step 5: Calculate the cosine of the angle between the lines

Using the formula for the cosine of the angle \(\theta\) between two vectors:
\[ \cos \theta = \frac{D_1 \cdot D_2}{|D_1| |D_2|} \]
Substituting the values we found:
\[ \cos \theta = \frac{0}{\sqrt{62} \cdot \sqrt{21}} = 0 \]

### Step 6: Determine the angle

Since \(\cos \theta = 0\), we find that:
\[ \theta = 90^\circ \]

### Step 7: Check if the lines are parallel or perpendicular

Since the angle between the lines is \(90^\circ\), we conclude that the lines are perpendicular.

### Final Answer

The angle between the lines is \(90^\circ\) and the lines are perpendicular.

---