Which of the following vector is perpendiuclar to the vector `vecA=2hati+3hatj+4hatk`?

Options

A. `hati+hatj+hatk`
B. `4hati+3hatj-2hatk`
C. `hati+3hatj+hatk`
D. `hati+2hatj-2hatk`

Answer

To determine which vector is perpendicular to the vector \(\vec{A} = 2\hat{i} + 3\hat{j} + 4\hat{k}\), we need to use the property that two vectors are perpendicular if their dot product is zero. We will check each option provided to see which one satisfies this condition.

### Step-by-Step Solution:

1. **Identify the Given Vector**:
The vector given is:
\[ \vec{A} = 2\hat{i} + 3\hat{j} + 4\hat{k} \]

2. **Understand the Dot Product**:
The dot product of two vectors \(\vec{A} = a_1\hat{i} + b_1\hat{j} + c_1\hat{k}\) and \(\vec{B} = a_2\hat{i} + b_2\hat{j} + c_2\hat{k}\) is given by:
\[ \vec{A} \cdot \vec{B} = a_1a_2 + b_1b_2 + c_1c_2 \]
For the vectors to be perpendicular, this dot product must equal zero.

3. **Check Each Option**:
We will evaluate each option provided to find the one that results in a dot product of zero.

- **Option 1**: \(\vec{B_1} = \hat{i} + \hat{j} + \hat{k}\)
\[ \vec{A} \cdot \vec{B_1} = (2)(1) + (3)(1) + (4)(1) = 2 + 3 + 4 = 9 \quad (\text{Not perpendicular}) \]

- **Option 2**: \(\vec{B_2} = 4\hat{i} + 3\hat{j} - 2\hat{k}\)
\[ \vec{A} \cdot \vec{B_2} = (2)(4) + (3)(3) + (4)(-2) = 8 + 9 - 8 = 9 \quad (\text{Not perpendicular}) \]

- **Option 3**: \(\vec{B_3} = \hat{i} + 3\hat{j} + \hat{k}\)
\[ \vec{A} \cdot \vec{B_3} = (2)(1) + (3)(3) + (4)(1) = 2 + 9 + 4 = 15 \quad (\text{Not perpendicular}) \]

- **Option 4**: \(\vec{B_4} = \hat{i} + 2\hat{j} - 2\hat{k}\)
\[ \vec{A} \cdot \vec{B_4} = (2)(1) + (3)(2) + (4)(-2) = 2 + 6 - 8 = 0 \quad (\text{Perpendicular}) \]

4. **Conclusion**:
The vector that is perpendicular to \(\vec{A}\) is:
\[ \vec{B_4} = \hat{i} + 2\hat{j} - 2\hat{k} \]