Find the directions ratios of the line perpendicular to the lines
` (x-7)/(2) = ( y+ 7)/(-3) =(z-6)/(1) and (x+5)/(1) = (y+3)/(2)= (z-6)/(-2)`

Answer

To find the direction ratios of the line that is perpendicular to the given lines, we can follow these steps:

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

The first line is given by the equation:
\[ \frac{x-7}{2} = \frac{y+7}{-3} = \frac{z-6}{1} \]
From this equation, we can extract the direction ratios (denoted as \(a_1, b_1, c_1\)):
- \(a_1 = 2\)
- \(b_1 = -3\)
- \(c_1 = 1\)

Thus, the direction ratios of the first line are:
\[ \text{Direction Ratios of Line 1} = (2, -3, 1) \]

The second line is given by the equation:
\[ \frac{x+5}{1} = \frac{y+3}{2} = \frac{z-6}{-2} \]
From this equation, we can extract the direction ratios (denoted as \(a_2, b_2, c_2\)):
- \(a_2 = 1\)
- \(b_2 = 2\)
- \(c_2 = -2\)

Thus, the direction ratios of the second line are:
\[ \text{Direction Ratios of Line 2} = (1, 2, -2) \]

### Step 2: Set up the cross product to find the direction ratios of the perpendicular line

To find the direction ratios of the line that is perpendicular to both lines, we need to calculate the cross product of the two direction ratio vectors:
\[ \mathbf{A} = (2, -3, 1) \quad \text{and} \quad \mathbf{B} = (1, 2, -2) \]

The cross product \(\mathbf{A} \times \mathbf{B}\) can be calculated using the determinant:
\[ \mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & -3 & 1 \\ 1 & 2 & -2 \end{vmatrix} \]

### Step 3: Calculate the determinant

Calculating the determinant, we have:
\[ \mathbf{A} \times \mathbf{B} = \mathbf{i} \begin{vmatrix} -3 & 1 \\ 2 & -2 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 2 & 1 \\ 1 & -2 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 2 & -3 \\ 1 & 2 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
1. For \(\mathbf{i}\):
\[ (-3)(-2) - (1)(2) = 6 - 2 = 4 \]

2. For \(\mathbf{j}\):
\[ (2)(-2) - (1)(1) = -4 - 1 = -5 \]

3. For \(\mathbf{k}\):
\[ (2)(2) - (-3)(1) = 4 + 3 = 7 \]

Putting it all together:
\[ \mathbf{A} \times \mathbf{B} = 4\mathbf{i} + 5\mathbf{j} + 7\mathbf{k} \]

### Step 4: Write the direction ratios

Thus, the direction ratios of the line that is perpendicular to both given lines are:
\[ (4, 5, 7) \]

### Final Answer:
The direction ratios of the line perpendicular to the given lines are:
\[ \boxed{(4, 5, 7)} \]

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