If `vecA=8veci-4hatj+8hatk` and `vecB=-4hati+2hatj+alphahatk` are perpendicular to each other then find value of `(alpha)` ?

Options

A. 0
B. `vec(0)`
C. `1`
D. `3`

Explanation

To find the value of \( \alpha \) such that the vectors \( \vec{A} = 8\hat{i} - 4\hat{j} + 8\hat{k} \) and \( \vec{B} = -4\hat{i} + 2\hat{j} + \alpha\hat{k} \) are perpendicular, we will use the property that the dot product of two perpendicular vectors is zero.

### Step 1: Set up the dot product equation
The dot product of two vectors \( \vec{A} \) and \( \vec{B} \) is given by:
\[ \vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z \]
where \( A_x, A_y, A_z \) are the components of \( \vec{A} \) and \( B_x, B_y, B_z \) are the components of \( \vec{B} \).

### Step 2: Substitute the components
From the given vectors:
- \( \vec{A} = 8\hat{i} - 4\hat{j} + 8\hat{k} \) implies \( A_x = 8, A_y = -4, A_z = 8 \)
- \( \vec{B} = -4\hat{i} + 2\hat{j} + \alpha\hat{k} \) implies \( B_x = -4, B_y = 2, B_z = \alpha \)

Now, substituting these values into the dot product equation:
\[ \vec{A} \cdot \vec{B} = (8)(-4) + (-4)(2) + (8)(\alpha) \]

### Step 3: Calculate the dot product
Calculating each term:
\[ \vec{A} \cdot \vec{B} = -32 - 8 + 8\alpha \]
This simplifies to:
\[ \vec{A} \cdot \vec{B} = -40 + 8\alpha \]

### Step 4: Set the dot product to zero
Since the vectors are perpendicular, we set the dot product equal to zero:
\[ -40 + 8\alpha = 0 \]

### Step 5: Solve for \( \alpha \)
Rearranging the equation gives:
\[ 8\alpha = 40 \]
Now, divide both sides by 8:
\[ \alpha = \frac{40}{8} = 5 \]

### Final Answer
Thus, the value of \( \alpha \) is \( 5 \).

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