Find value of `lambda` if two lines having direction ratios 1, -2, 3 and `2- lambda , 1+ lambda, - lambda, ` are perpendicular
to each other

Options

A. 1
B. `-1`
C. 0
D. 2

Answer

To find the value of \( \lambda \) such that the two lines with direction ratios \( (1, -2, 3) \) and \( (2 - \lambda, 1 + \lambda, -\lambda) \) are perpendicular, we can follow these steps:

### Step 1: Understand the Condition for Perpendicularity
Two vectors (or lines) are perpendicular if the dot product of their direction ratios is zero. For direction ratios \( (A_1, B_1, C_1) \) and \( (A_2, B_2, C_2) \), the condition is:
\[ A_1 \cdot A_2 + B_1 \cdot B_2 + C_1 \cdot C_2 = 0 \]

### Step 2: Identify the Direction Ratios
From the problem, we have:
- First line direction ratios: \( (1, -2, 3) \)
- Second line direction ratios: \( (2 - \lambda, 1 + \lambda, -\lambda) \)

### Step 3: Set Up the Dot Product Equation
Using the direction ratios, we can set up the equation:
\[ 1 \cdot (2 - \lambda) + (-2) \cdot (1 + \lambda) + 3 \cdot (-\lambda) = 0 \]

### Step 4: Simplify the Equation
Now, we simplify the equation:
\[ (2 - \lambda) - 2(1 + \lambda) - 3\lambda = 0 \]
Expanding this gives:
\[ 2 - \lambda - 2 - 2\lambda - 3\lambda = 0 \]
Combining like terms results in:
\[ 2 - 2 - \lambda - 2\lambda - 3\lambda = 0 \]
This simplifies to:
\[ -6\lambda = 0 \]

### Step 5: Solve for \( \lambda \)
From the equation \( -6\lambda = 0 \), we can solve for \( \lambda \):
\[ \lambda = 0 \]

### Conclusion
The value of \( \lambda \) is \( 0 \).

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