If a vector `2 hat (i) + 3 hat(j) + 8 hat(k)` is perpendicular to the vector `- 4 hat(i) + 4 hat(j) + alpha(k)`. Then the value of `alpha` is

Explanation

To solve the problem, we need to determine 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. **Understand the Condition for Perpendicular Vectors**:
Two vectors are perpendicular if their dot product is zero. Therefore, we need to set up the equation:
\[ \mathbf{A} \cdot \mathbf{B} = 0 \]

2. **Write the Dot Product**:
The dot product \( \mathbf{A} \cdot \mathbf{B} \) can be calculated as follows:
\[ \mathbf{A} \cdot \mathbf{B} = (2\hat{i} + 3\hat{j} + 8\hat{k}) \cdot (-4\hat{i} + 4\hat{j} + \alpha\hat{k}) \]

3. **Calculate the Dot Product**:
Using the properties of dot products, we multiply the corresponding components:
\[ \mathbf{A} \cdot \mathbf{B} = (2 \cdot -4) + (3 \cdot 4) + (8 \cdot \alpha) \]
Simplifying this gives:
\[ -8 + 12 + 8\alpha = 0 \]

4. **Simplify the Equation**:
Combine the constant terms:
\[ 4 + 8\alpha = 0 \]

5. **Solve for Alpha**:
Rearranging the equation to isolate \( \alpha \):
\[ 8\alpha = -4 \]
Dividing both sides by 8:
\[ \alpha = -\frac{4}{8} = -\frac{1}{2} \]

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

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