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

Options

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

Answer

To find the value of \( \lambda \) such that the vectors \( \vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k} \) and \( \vec{b} = 3\hat{i} + 2\hat{j} - \lambda\hat{k} \) are perpendicular, we will use the property that the dot product of two perpendicular vectors is zero.

### Step-by-Step Solution:

1. **Write down the dot product formula**:
The dot product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by:
\[ \vec{a} \cdot \vec{b} = (a_1 b_1 + a_2 b_2 + a_3 b_3) \]
where \( a_1, a_2, a_3 \) are the components of \( \vec{a} \) and \( b_1, b_2, b_3 \) are the components of \( \vec{b} \).

2. **Substitute the components of the vectors**:
For our vectors:
- \( \vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k} \) has components \( (2, 3, 4) \)
- \( \vec{b} = 3\hat{i} + 2\hat{j} - \lambda\hat{k} \) has components \( (3, 2, -\lambda) \)

Thus, the dot product becomes:
\[ \vec{a} \cdot \vec{b} = 2 \cdot 3 + 3 \cdot 2 + 4 \cdot (-\lambda) \]

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

4. **Set the dot product equal to zero**:
Since the vectors are perpendicular, we set the dot product to zero:
\[ 12 - 4\lambda = 0 \]

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

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