If `vecA= 4hati-2hatj+4hatk and vecB= -4hati+2hatj+alphahatk` are perpendicular to each other then find value of `alpha` ?

Explanation

To find the value of \( \alpha \) such that the vectors \( \vec{A} \) and \( \vec{B} \) are perpendicular, we can follow these steps:

### Step 1: Understand the Condition for Perpendicular Vectors
Two vectors are perpendicular if their dot product is equal to zero. Therefore, we need to calculate the dot product of \( \vec{A} \) and \( \vec{B} \) and set it equal to zero.

### Step 2: Write Down the Given Vectors
We have:
\[ \vec{A} = 4\hat{i} - 2\hat{j} + 4\hat{k} \]
\[ \vec{B} = -4\hat{i} + 2\hat{j} + \alpha\hat{k} \]

### Step 3: Calculate the Dot Product
The dot product \( \vec{A} \cdot \vec{B} \) is calculated as follows:
\[ \vec{A} \cdot \vec{B} = (4)(-4) + (-2)(2) + (4)(\alpha) \]
Calculating each term:
- The first term: \( 4 \times -4 = -16 \)
- The second term: \( -2 \times 2 = -4 \)
- The third term: \( 4 \times \alpha = 4\alpha \)

Putting it all together:
\[ \vec{A} \cdot \vec{B} = -16 - 4 + 4\alpha \]
This simplifies to:
\[ \vec{A} \cdot \vec{B} = -20 + 4\alpha \]

### Step 4: Set the Dot Product Equal to Zero
Since the vectors are perpendicular, we set the dot product to zero:
\[ -20 + 4\alpha = 0 \]

### Step 5: Solve for \( \alpha \)
Rearranging the equation gives:
\[ 4\alpha = 20 \]
Dividing both sides by 4:
\[ \alpha = 5 \]

### Final Answer
The value of \( \alpha \) is \( 5 \).

---