Which of the following vector is perpendicular to the vector ` vec A = 2 hat i + 3 hat j + 4 hat k `?

Answer

To determine which of the given vectors is perpendicular to the vector \( \vec{A} = 2\hat{i} + 3\hat{j} + 4\hat{k} \), we can use the property of the dot product. Two vectors \( \vec{A} \) and \( \vec{B} \) are perpendicular if their dot product is zero, i.e.,

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

### Step-by-Step Solution:

1. **Identify the Vector \( \vec{A} \)**:
\[ \vec{A} = 2\hat{i} + 3\hat{j} + 4\hat{k} \]

2. **Calculate the Dot Product**:
We need to check the dot product of \( \vec{A} \) with each of the given options. Let's denote the options as \( \vec{B_1}, \vec{B_2}, \vec{B_3}, \vec{B_4} \).

3. **Option 1: \( \vec{B_1} = 3\hat{i} + 2\hat{j} + 3\hat{k} \)**:
\[ \vec{A} \cdot \vec{B_1} = (2)(3) + (3)(2) + (4)(3) = 6 + 6 + 12 = 24 \]
Since \( 24 \neq 0 \), this vector is not perpendicular.

4. **Option 2: \( \vec{B_2} = 3\hat{i} + 2\hat{j} - 3\hat{k} \)**:
\[ \vec{A} \cdot \vec{B_2} = (2)(3) + (3)(2) + (4)(-3) = 6 + 6 - 12 = 0 \]
Since \( 0 = 0 \), this vector is perpendicular.

5. **Option 3: \( \vec{B_3} = 1\hat{i} + 1\hat{j} + 1\hat{k} \)**:
\[ \vec{A} \cdot \vec{B_3} = (2)(1) + (3)(1) + (4)(1) = 2 + 3 + 4 = 9 \]
Since \( 9 \neq 0 \), this vector is not perpendicular.

6. **Option 4: \( \vec{B_4} = 0\hat{i} + 0\hat{j} + 0\hat{k} \)**:
\[ \vec{A} \cdot \vec{B_4} = (2)(0) + (3)(0) + (4)(0) = 0 \]
While this is technically perpendicular, it is the zero vector, which is not a valid option in most contexts.

### Conclusion:
The vector that is perpendicular to \( \vec{A} \) is:
\[ \vec{B_2} = 3\hat{i} + 2\hat{j} - 3\hat{k} \]