The vector `2i+j-k` is perpendicular to `i-4j+lambdak` if `lambda` is equal to

Options

A. 0
B. `-1`
C. `-2`
D. `-3`

Answer

To determine the value of \( \lambda \) such that the vector \( 2\mathbf{i} + \mathbf{j} - \mathbf{k} \) is perpendicular to the vector \( \mathbf{i} - 4\mathbf{j} + \lambda \mathbf{k} \), we can use the property that two vectors are perpendicular if their dot product is zero.

### Step-by-Step Solution:

1. **Identify the vectors**:
Let \( \mathbf{A} = 2\mathbf{i} + \mathbf{j} - \mathbf{k} \) and \( \mathbf{B} = \mathbf{i} - 4\mathbf{j} + \lambda \mathbf{k} \).

2. **Set up the dot product**:
The dot product \( \mathbf{A} \cdot \mathbf{B} \) can be calculated as follows:
\[ \mathbf{A} \cdot \mathbf{B} = (2\mathbf{i} + \mathbf{j} - \mathbf{k}) \cdot (\mathbf{i} - 4\mathbf{j} + \lambda \mathbf{k}) \]

3. **Calculate the dot product**:
Using the distributive property of the dot product:
\[ \mathbf{A} \cdot \mathbf{B} = 2 \cdot 1 + 1 \cdot (-4) + (-1) \cdot \lambda \]
This simplifies to:
\[ \mathbf{A} \cdot \mathbf{B} = 2 - 4 - \lambda \]

4. **Set the dot product to zero**:
Since the vectors are perpendicular, we set the dot product equal to zero:
\[ 2 - 4 - \lambda = 0 \]

5. **Solve for \( \lambda \)**:
Rearranging the equation gives:
\[ -2 - \lambda = 0 \implies \lambda = -2 \]

### Final Answer:
Thus, the value of \( \lambda \) is \( -2 \).