The value of `lambda` for which the vectors `2 hat(i) + lambda hat(j) + hat(k)` and `hat(i) + 2 hat(j) + 3 hat(k)` are perpendicular is

Options

A. 0
B. 1
C. `(3)/(2)`
D. `- ( 5)/(2)`

Answer

To find the value of \( \lambda \) for which the vectors \( \mathbf{A} = 2\hat{i} + \lambda\hat{j} + \hat{k} \) and \( \mathbf{B} = \hat{i} + 2\hat{j} + 3\hat{k} \) are perpendicular, we can follow these steps:

### Step 1: Understand the condition for perpendicular vectors
Two vectors are perpendicular if their dot product is zero. Therefore, we need to set up the equation:

\[ \mathbf{A} \cdot \mathbf{B} = 0 \]

### Step 2: Write down the vectors
Let:
\[ \mathbf{A} = 2\hat{i} + \lambda\hat{j} + \hat{k} \]
\[ \mathbf{B} = \hat{i} + 2\hat{j} + 3\hat{k} \]

### Step 3: Calculate the dot product
Now, we compute the dot product \( \mathbf{A} \cdot \mathbf{B} \):

\[ \mathbf{A} \cdot \mathbf{B} = (2\hat{i} + \lambda\hat{j} + \hat{k}) \cdot (\hat{i} + 2\hat{j} + 3\hat{k}) \]

Using the properties of dot product:
\[ = 2(\hat{i} \cdot \hat{i}) + \lambda(\hat{j} \cdot 2\hat{j}) + 1(\hat{k} \cdot 3\hat{k}) \]

### Step 4: Simplify the dot product
Calculating each term:
- \( \hat{i} \cdot \hat{i} = 1 \)
- \( \hat{j} \cdot \hat{j} = 1 \)
- \( \hat{k} \cdot \hat{k} = 1 \)

Thus:
\[ = 2(1) + \lambda(2) + 1(3) = 2 + 2\lambda + 3 \]

### Step 5: Set the dot product equal to zero
Now, set the dot product equal to zero:
\[ 2 + 2\lambda + 3 = 0 \]

### Step 6: Solve for \( \lambda \)
Combine the constants:
\[ 5 + 2\lambda = 0 \]
\[ 2\lambda = -5 \]
\[ \lambda = -\frac{5}{2} \]

### Conclusion
The value of \( \lambda \) for which the vectors are perpendicular is:
\[ \lambda = -\frac{5}{2} \]