What is the projection of (`3hati+3hatj+8hatk)` is perpendicular to the vector `(4hatj-4hati+alpha hatk)`, then the value of `alpha` is :

Answer

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

### Step-by-Step Solution:

1. **Identify the vectors**:
- Let \( \vec{A} = 3\hat{i} + 3\hat{j} + 8\hat{k} \)
- Let \( \vec{B} = -4\hat{i} + 4\hat{j} + \alpha\hat{k} \)

2. **Use the property of perpendicular vectors**:
- Two vectors are perpendicular if their dot product is zero:
\[ \vec{A} \cdot \vec{B} = 0 \]

3. **Calculate the dot product**:
- The dot product \( \vec{A} \cdot \vec{B} \) is calculated as follows:
\[ \vec{A} \cdot \vec{B} = (3\hat{i} + 3\hat{j} + 8\hat{k}) \cdot (-4\hat{i} + 4\hat{j} + \alpha\hat{k}) \]
- This expands to:
\[ = 3(-4) + 3(4) + 8(\alpha) \]
- Simplifying this gives:
\[ = -12 + 12 + 8\alpha \]

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

5. **Solve for \( \alpha \)**:
- Dividing both sides by 8 gives:
\[ \alpha = 0 \]

### Final Answer:
The value of \( \alpha \) is \( 0 \).