Two vector `vec(A) = 2 hat(i) + 3hat(j) - 4 hat(k) and vec(B) = 4hat(i) + 8 hat(j) + x hat(k)` are such that he component of `vec(B) " along " vec(A)` is zero. Then the value of x will be:

Options

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

Answer

To solve the problem, we need to determine the value of \( x \) such that the component of vector \( \vec{B} \) along vector \( \vec{A} \) is zero. This implies that the two vectors are perpendicular, and therefore their dot product must be zero.

### Step-by-Step Solution:

1. **Identify the Vectors**:
- Given vectors are:
\[ \vec{A} = 2\hat{i} + 3\hat{j} - 4\hat{k} \]
\[ \vec{B} = 4\hat{i} + 8\hat{j} + x\hat{k} \]

2. **Set Up the Dot Product**:
- The dot product of two vectors \( \vec{A} \) and \( \vec{B} \) is given by:
\[ \vec{A} \cdot \vec{B} = (2)(4) + (3)(8) + (-4)(x) \]
- This simplifies to:
\[ \vec{A} \cdot \vec{B} = 8 + 24 - 4x \]

3. **Set the Dot Product to Zero**:
- Since the component of \( \vec{B} \) along \( \vec{A} \) is zero, we set the dot product equal to zero:
\[ 8 + 24 - 4x = 0 \]

4. **Solve for \( x \)**:
- Combine like terms:
\[ 32 - 4x = 0 \]
- Rearranging gives:
\[ 4x = 32 \]
- Dividing both sides by 4:
\[ x = 8 \]

5. **Conclusion**:
- The value of \( x \) is:
\[ \boxed{8} \]