If a vector `2hat(i)+3hat(j)+8hat(k)` is perpendicular to the vector `4hat(j)-4hat(i)+alpha hat(k)`. Then the value of `alpha` is

Options

A. -1
B. `(1)/(2)`
C. `-(1)/(2)`
D. 1

Answer

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 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 vectors \( \mathbf{A} \) and \( \mathbf{B} \) and set it equal to zero.

### Step 2: Write down the vectors
Let:
- \( \mathbf{A} = 2\hat{i} + 3\hat{j} + 8\hat{k} \)
- \( \mathbf{B} = -4\hat{i} + 4\hat{j} + \alpha\hat{k} \)

### Step 3: Calculate the dot product
The dot product \( \mathbf{A} \cdot \mathbf{B} \) is calculated as follows:
\[ \mathbf{A} \cdot \mathbf{B} = (2)(-4) + (3)(4) + (8)(\alpha) \]

### Step 4: Simplify the dot product
Now, we can simplify the expression:
\[ \mathbf{A} \cdot \mathbf{B} = -8 + 12 + 8\alpha \]
\[ \mathbf{A} \cdot \mathbf{B} = 4 + 8\alpha \]

### Step 5: Set the dot product equal to zero
Since the vectors are perpendicular, we set the dot product to zero:
\[ 4 + 8\alpha = 0 \]

### Step 6: Solve for \( \alpha \)
Now, we can solve for \( \alpha \):
\[ 8\alpha = -4 \]
\[ \alpha = -\frac{4}{8} = -\frac{1}{2} \]

### Final Answer
Thus, the value of \( \alpha \) is \( -\frac{1}{2} \).

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