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

Options

A. `1//2`
B. `-1`
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 compute the dot product of vectors \( \mathbf{A} \) and \( \mathbf{B} \) and set it equal to zero.

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

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

Using the properties of the dot product:
\[ \mathbf{A} \cdot \mathbf{B} = (2 \cdot 4) + (3 \cdot -4) + (8 \cdot \alpha) \]
\[ = 8 - 12 + 8\alpha \]

4. **Set the Dot Product to Zero**:
Since the vectors are perpendicular, we set the dot product equal to zero:
\[ 8 - 12 + 8\alpha = 0 \]

5. **Simplify the Equation**:
Simplifying the equation gives:
\[ -4 + 8\alpha = 0 \]

6. **Solve for \( \alpha \)**:
Rearranging the equation to solve for \( \alpha \):
\[ 8\alpha = 4 \]
\[ \alpha = \frac{4}{8} = \frac{1}{2} \]

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

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