If a vector ` 2 hati+ 3 hatj+ 8 hatk ` is perpendicular to the vector ` 4 hatj-4hai+ alpha hatk` , then the value of `alpha`

Options

A. `-1`
B. `-(1)/(2)`
C. `(1)/(2)`
D. `1`

Explanation

To solve the problem, we need to find the value of \( \alpha \) such that the vector \( \mathbf{A} = 2\hat{i} + 3\hat{j} + 8\hat{k} \) is perpendicular to the vector \( \mathbf{B} = -4\hat{i} + 4\hat{j} + \alpha\hat{k} \).

### Step-by-Step Solution:

1. **Understanding Perpendicular Vectors**:
Two vectors are perpendicular if their dot product is zero. Therefore, we need to calculate the dot product \( \mathbf{A} \cdot \mathbf{B} \) and set it equal to zero.

2. **Write Down the Vectors**:
\[ \mathbf{A} = 2\hat{i} + 3\hat{j} + 8\hat{k} \]
\[ \mathbf{B} = -4\hat{i} + 4\hat{j} + \alpha\hat{k} \]

3. **Calculate the Dot Product**:
The dot product \( \mathbf{A} \cdot \mathbf{B} \) is calculated as follows:
\[ \mathbf{A} \cdot \mathbf{B} = (2)(-4) + (3)(4) + (8)(\alpha) \]

4. **Perform the Multiplications**:
\[ \mathbf{A} \cdot \mathbf{B} = -8 + 12 + 8\alpha \]

5. **Combine Like Terms**:
\[ \mathbf{A} \cdot \mathbf{B} = 4 + 8\alpha \]

6. **Set the Dot Product Equal to Zero**:
Since the vectors are perpendicular:
\[ 4 + 8\alpha = 0 \]

7. **Solve for \( \alpha \)**:
\[ 8\alpha = -4 \]
\[ \alpha = -\frac{4}{8} = -\frac{1}{2} \]

### Final Answer:
The value of \( \alpha \) is \( -\frac{1}{2} \).