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

Options

A. 2
B. 4
C. 6
D. 8

Answer

To find the value of \( \lambda \) for which the two vectors \( \mathbf{a} = 2\hat{i} - \hat{j} + 2\hat{k} \) and \( \mathbf{b} = 3\hat{i} + \lambda \hat{j} + \hat{k} \) 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 find \( \mathbf{a} \cdot \mathbf{b} = 0 \).

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

### Step 3: Calculate the dot product
The dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as follows:
\[ \mathbf{a} \cdot \mathbf{b} = (2)(3) + (-1)(\lambda) + (2)(1) \]
This simplifies to:
\[ \mathbf{a} \cdot \mathbf{b} = 6 - \lambda + 2 \]
\[ \mathbf{a} \cdot \mathbf{b} = 8 - \lambda \]

### Step 4: Set the dot product to zero
Since the vectors are perpendicular, we set the dot product equal to zero:
\[ 8 - \lambda = 0 \]

### Step 5: Solve for \( \lambda \)
Now, we solve for \( \lambda \):
\[ \lambda = 8 \]

### Final Answer
The value of \( \lambda \) for which the two vectors are perpendicular is:
\[ \lambda = 8 \]

---