What is the value of `alpha`? If a vector `2hati+3hatj+2hatk` is perpendicular to the vector `-3hati+8hatj+alphahatk`.

Explanation

To find the value of `alpha` such that the vector \( \vec{A} = 2\hat{i} + 3\hat{j} + 2\hat{k} \) is perpendicular to the vector \( \vec{B} = -3\hat{i} + 8\hat{j} + \alpha\hat{k} \), we can use the property that two vectors are perpendicular if their dot product is zero.

### Step-by-step Solution:

1. **Write down the dot product condition:**
\[ \vec{A} \cdot \vec{B} = 0 \]

2. **Substitute the vectors into the dot product:**
\[ (2\hat{i} + 3\hat{j} + 2\hat{k}) \cdot (-3\hat{i} + 8\hat{j} + \alpha\hat{k}) = 0 \]

3. **Calculate the dot product:**
- The dot product is calculated as follows:
\[ \vec{A} \cdot \vec{B} = (2)(-3) + (3)(8) + (2)(\alpha) \]
- This simplifies to:
\[ -6 + 24 + 2\alpha = 0 \]

4. **Combine like terms:**
\[ 18 + 2\alpha = 0 \]

5. **Isolate \( \alpha \):**
\[ 2\alpha = -18 \]
\[ \alpha = -9 \]

### Final Answer:
Thus, the value of \( \alpha \) is \( -9 \).

---