If `vec(a) = 2 hat(i) + 3 hat (j) + 4 hat(k) and vec(b) = 3 hat(i) + 2 hat(j) - lambda hat(k)` are perpendicular, then what is the value of `lambda` ?

Options

A. 2
B. 3
C. 4
D. 5

Explanation

To solve the problem, we need to find the value of \( \lambda \) such that the vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular. Two vectors are perpendicular if their dot product is zero.

### Step-by-Step Solution:

1. **Identify the vectors:**
\[ \vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \]
\[ \vec{b} = 3 \hat{i} + 2 \hat{j} - \lambda \hat{k} \]

2. **Set up the dot product:**
The dot product \( \vec{a} \cdot \vec{b} \) is given by:
\[ \vec{a} \cdot \vec{b} = (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) \cdot (3 \hat{i} + 2 \hat{j} - \lambda \hat{k}) \]

3. **Calculate the dot product:**
Using the formula for the dot product:
\[ \vec{a} \cdot \vec{b} = (2 \cdot 3) + (3 \cdot 2) + (4 \cdot -\lambda) \]
Simplifying this gives:
\[ \vec{a} \cdot \vec{b} = 6 + 6 - 4\lambda \]
\[ \vec{a} \cdot \vec{b} = 12 - 4\lambda \]

4. **Set the dot product to zero:**
Since \( \vec{a} \) and \( \vec{b} \) are perpendicular, we set the dot product equal to zero:
\[ 12 - 4\lambda = 0 \]

5. **Solve for \( \lambda \):**
Rearranging the equation:
\[ 4\lambda = 12 \]
Dividing both sides by 4:
\[ \lambda = 3 \]

### Final Answer:
The value of \( \lambda \) is \( 3 \).