Direction rations of the line which is perpendicular to the lines with direction ratios `-1,2,2 and 0,2,1` are

Options

A. `1,1,2`
B. `2,-1,2`
C. `-2,1,2`
D. `2,1,-2`

Answer

To find the direction ratios of the line that is perpendicular to the two given lines with direction ratios \((-1, 2, 2)\) and \((0, 2, 1)\), we can follow these steps:

### Step 1: Understand the Problem
We are given two lines with direction ratios:
- Line 1 (L1): \((-1, 2, 2)\)
- Line 2 (L2): \((0, 2, 1)\)

We need to find the direction ratios of a line (L3) that is perpendicular to both L1 and L2.

### Step 2: Set Up the Scalar Product Equations
For two lines to be perpendicular, the scalar product of their direction ratios must equal zero.

Let the direction ratios of line L3 be \((A, B, C)\).

1. For L1 and L3:
\[ (-1)A + 2B + 2C = 0 \quad \text{(Equation 1)} \]

2. For L2 and L3:
\[ 0A + 2B + 1C = 0 \quad \text{(Equation 2)} \]

### Step 3: Solve the Equations
From Equation 2, we can express \(C\) in terms of \(B\):
\[ 2B + C = 0 \implies C = -2B \quad \text{(Substituting this into Equation 1)} \]

Now substitute \(C\) in Equation 1:
\[ -1A + 2B + 2(-2B) = 0 \]
This simplifies to:
\[ -A + 2B - 4B = 0 \implies -A - 2B = 0 \implies A = -2B \quad \text{(Equation 3)} \]

### Step 4: Express A, B, C in Terms of B
Now we have:
- From Equation 3: \(A = -2B\)
- From the expression for \(C\): \(C = -2B\)

Thus, we can express the direction ratios in terms of \(B\):
\[ A = -2B, \quad B = B, \quad C = -2B \]

### Step 5: Choose a Value for B
To find specific direction ratios, we can choose \(B = 1\):
\[ A = -2(1) = -2, \quad B = 1, \quad C = -2(1) = -2 \]
Thus, the direction ratios of line L3 are:
\[ (-2, 1, -2) \]

### Final Answer
The direction ratios of the line that is perpendicular to the given lines are:
\[ \boxed{(-2, 1, -2)} \]