If the vectors`2i + j + kand i -4j + lambdak` are perpendicular,then`lambda =`

Options

A. 4
B. `-5`
C. 2
D. 1

Answer

To solve the problem, we need to determine the value of \( \lambda \) such that the vectors \( \mathbf{A} = 2\mathbf{i} + \mathbf{j} + \mathbf{k} \) and \( \mathbf{B} = \mathbf{i} - 4\mathbf{j} + \lambda \mathbf{k} \) are perpendicular.

### Step-by-Step Solution:

1. **Understanding the Condition for Perpendicular Vectors**:
Two vectors are perpendicular if their dot product is equal to zero. Therefore, we need to find the dot product of the two given vectors and set it equal to zero.

2. **Write the Vectors**:
- Let \( \mathbf{A} = 2\mathbf{i} + \mathbf{j} + \mathbf{k} \)
- Let \( \mathbf{B} = \mathbf{i} - 4\mathbf{j} + \lambda \mathbf{k} \)

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

Using the properties of dot product:
\[ \mathbf{A} \cdot \mathbf{B} = (2)(1) + (1)(-4) + (1)(\lambda) \]

Simplifying this gives:
\[ \mathbf{A} \cdot \mathbf{B} = 2 - 4 + \lambda = \lambda - 2 \]

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

5. **Solve for \( \lambda \)**:
Solving the equation gives:
\[ \lambda = 2 \]

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

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