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 concept of the dot product. Two vectors are perpendicular if their dot product is equal to zero.

### Step-by-Step Solution:

1. **Understand the Dot Product**: The dot product of two vectors \(\vec{A} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\) and \(\vec{B} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}\) is given by:
\[ \vec{A} \cdot \vec{B} = a_1b_1 + a_2b_2 + a_3b_3 \]
For \(\vec{A}\) and \(\vec{B}\) to be perpendicular, we need:
\[ \vec{A} \cdot \vec{B} = 0 \]

2. **Check Each Option**: We will check the dot product of \(\vec{A}\) with each of the given vectors (options).

- **Option A**: \(\vec{B} = \hat{i} + 4\hat{k}\)
\[ \vec{A} \cdot \vec{B} = (2)(1) + (3)(0) + (4)(4) = 2 + 0 + 16 = 18 \quad (\text{Not perpendicular}) \]

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

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

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

3. **Conclusion**: The vector that is perpendicular to \(\vec{A}\) is option D: \(\hat{i} + 2\hat{j} - 2\hat{k}\).