Two vector `vec(a)=3hat(i)+8hat(j)-2hat(k)` and `vec(b)=6hat(i)+16hat(j)+xhat(k)` are such that the component of `vec(b)`perpendicular to `vec(a)` is zero. Then the value of `x` will be `:-`

Options

A. `8`
B. `-4`
C. `+4`
D. `-8`

Explanation

To solve the problem, we need to find the value of \( x \) such that the vector \( \vec{b} \) is parallel to the vector \( \vec{a} \). This means that the two vectors have the same direction, which can be expressed mathematically as:

\[ \vec{b} = \lambda \vec{a} \]

where \( \lambda \) is a scalar.

### Step 1: Write down the vectors
Given:
\[ \vec{a} = 3\hat{i} + 8\hat{j} - 2\hat{k} \]
\[ \vec{b} = 6\hat{i} + 16\hat{j} + x\hat{k} \]

### Step 2: Set up the equation for parallel vectors
Since \( \vec{b} \) is parallel to \( \vec{a} \), we can express this as:
\[ 6\hat{i} + 16\hat{j} + x\hat{k} = \lambda (3\hat{i} + 8\hat{j} - 2\hat{k}) \]

### Step 3: Equate the components
Now, we can equate the coefficients of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) from both sides:

1. For \( \hat{i} \):
\[ 6 = 3\lambda \quad \text{(1)} \]

2. For \( \hat{j} \):
\[ 16 = 8\lambda \quad \text{(2)} \]

3. For \( \hat{k} \):
\[ x = -2\lambda \quad \text{(3)} \]

### Step 4: Solve for \( \lambda \)
From equation (1):
\[ \lambda = \frac{6}{3} = 2 \]

### Step 5: Verify \( \lambda \) with equation (2)
Substituting \( \lambda = 2 \) into equation (2):
\[ 16 = 8 \times 2 \]
This is correct.

### Step 6: Find \( x \)
Now substitute \( \lambda = 2 \) into equation (3):
\[ x = -2\lambda = -2 \times 2 = -4 \]

### Conclusion
The value of \( x \) is:
\[ \boxed{-4} \]