The directionratios of the line which is perpendicular to the lines `(x-7)/2=(y+17)/(-3)=z-6 and x+5=(y+3)/2=(z-4)/(-2)` are (A) (4,5,7) (B) (4,-5,7) (C) (4,-5,-7) (D) (-4,5,7)

Options

A. 4,5,7
B. 4,-5,7
C. 4,-5,7
D. `-4,5,7`

Explanation

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

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

The first line is given by the equation:
\[ \frac{x-7}{2} = \frac{y+17}{-3} = z-6 \]
From this, we can extract the direction ratios (DR) of the first line, denoted as \( b_1 \):
- The direction ratios are \( (2, -3, 1) \).

The second line is given by the equation:
\[ x + 5 = \frac{y + 3}{2} = \frac{z - 4}{-2} \]
From this, we can extract the direction ratios of the second line, denoted as \( b_2 \):
- The direction ratios are \( (1, 2, -2) \).

### Step 2: Use the cross product to find the direction ratios of the line perpendicular to both lines.

To find the direction ratios of the line that is perpendicular to both lines, we will calculate the cross product of the vectors \( b_1 \) and \( b_2 \):
\[ b_1 = (2, -3, 1), \quad b_2 = (1, 2, -2) \]

The cross product \( b = b_1 \times b_2 \) can be computed using the determinant:
\[ b = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -3 & 1 \\ 1 & 2 & -2 \end{vmatrix} \]

### Step 3: Calculate the determinant.

Calculating the determinant:
\[ b = \hat{i} \begin{vmatrix} -3 & 1 \\ 2 & -2 \end{vmatrix} - \hat{j} \begin{vmatrix} 2 & 1 \\ 1 & -2 \end{vmatrix} + \hat{k} \begin{vmatrix} 2 & -3 \\ 1 & 2 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
1. For \( \hat{i} \):
\[ (-3)(-2) - (1)(2) = 6 - 2 = 4 \]
2. For \( \hat{j} \):
\[ (2)(-2) - (1)(1) = -4 - 1 = -5 \quad \text{(note the negative sign in front)} \]
Thus, we have \( +5 \hat{j} \).
3. For \( \hat{k} \):
\[ (2)(2) - (-3)(1) = 4 + 3 = 7 \]

Putting it all together:
\[ b = 4 \hat{i} + 5 \hat{j} + 7 \hat{k} \]

### Step 4: Write the final result.

The direction ratios of the required line are:
\[ (4, 5, 7) \]

### Conclusion

Thus, the correct option is (A) \( (4, 5, 7) \).

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