A vector `vecF_(1)` is along the positive `X`-axis. If its vectors product with another vector `vecF_(2)` is zero then `vecF_(2)` could be

Options

A. `4hatj`
B. `-(hati+hatj)`
C. `(hatj+hatk)`
D. `(-4hati)`

Answer

To solve the problem, we need to analyze the conditions under which the vector product (cross product) of two vectors is zero.

### Step-by-Step Solution:

1. **Understanding the Given Vectors**:
- We have a vector \( \vec{F_1} \) which is along the positive X-axis. We can represent this vector as:
\[ \vec{F_1} = F_1 \hat{i} \]
where \( F_1 \) is the magnitude of the vector and \( \hat{i} \) is the unit vector in the X direction.

2. **Condition for Zero Vector Product**:
- The vector product (cross product) of two vectors \( \vec{A} \) and \( \vec{B} \) is given by:
\[ \vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin \theta \hat{n} \]
where \( \theta \) is the angle between the two vectors and \( \hat{n} \) is the unit vector perpendicular to the plane formed by \( \vec{A} \) and \( \vec{B} \).

3. **Setting Up the Equation**:
- For the vector product \( \vec{F_1} \times \vec{F_2} \) to be zero, we need:
\[ |\vec{F_1}| |\vec{F_2}| \sin \theta = 0 \]
This can happen if either:
- \( |\vec{F_1}| = 0 \) (which is not the case since \( \vec{F_1} \) is given)
- \( |\vec{F_2}| = 0 \) (which means \( \vec{F_2} \) is a zero vector)
- \( \sin \theta = 0 \) (which means \( \theta = 0^\circ \) or \( \theta = 180^\circ \))

4. **Analyzing the Angles**:
- If \( \theta = 0^\circ \), it means \( \vec{F_2} \) is in the same direction as \( \vec{F_1} \) (parallel).
- If \( \theta = 180^\circ \), it means \( \vec{F_2} \) is in the opposite direction to \( \vec{F_1} \) (antiparallel).

5. **Conclusion**:
- Therefore, \( \vec{F_2} \) could either be parallel or antiparallel to \( \vec{F_1} \). This means \( \vec{F_2} \) could be any vector along the X-axis, including:
\[ \vec{F_2} = k \hat{i} \quad \text{for any scalar } k \]
- Specifically, \( k \) could be positive (for parallel) or negative (for antiparallel).

### Final Answer:
Thus, \( \vec{F_2} \) could be any vector along the X-axis, including \( \vec{F_2} = k \hat{i} \) where \( k \) is any real number.