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

Explanation

To solve the problem step by step, let's analyze the given information and apply the concepts of vectors and their properties.

### Step 1: Understand the Given Vectors
We are given that vector \(\vec{F_1}\) is along the positive X-axis. We can represent this vector as:
\[ \vec{F_1} = a \hat{i} \]
where \(a\) is the magnitude of the vector and \(\hat{i}\) is the unit vector in the direction of the X-axis.

### Step 2: Vector Product Condition
We know that the vector product (cross product) of \(\vec{F_1}\) and another vector \(\vec{F_2}\) is zero:
\[ \vec{F_1} \times \vec{F_2} = 0 \]
The cross product of two vectors is zero if and only if the vectors are parallel or one of the vectors is a zero vector.

### Step 3: Express the Cross Product
The magnitude of the cross product can be expressed as:
\[ |\vec{F_1} \times \vec{F_2}| = |\vec{F_1}| |\vec{F_2}| \sin \theta \]
where \(\theta\) is the angle between the two vectors. Since the cross product is zero, we can write:
\[ |\vec{F_1}| |\vec{F_2}| \sin \theta = 0 \]

### Step 4: Analyze the Conditions
For the product to be zero, either:
1. \(|\vec{F_1}| = 0\) (which is not the case since \(\vec{F_1}\) has a magnitude \(a > 0\)), or
2. \(|\vec{F_2}| = 0\) (which means \(\vec{F_2}\) is the zero vector), or
3. \(\sin \theta = 0\) (which implies \(\theta = 0^\circ\) or \(\theta = 180^\circ\)).

### Step 5: Determine the Implications of \(\theta\)
If \(\theta = 0^\circ\), then \(\vec{F_2}\) is in the same direction as \(\vec{F_1}\), meaning:
\[ \vec{F_2} = b \hat{i} \]
for some scalar \(b\) (where \(b\) can be positive or negative).

If \(\theta = 180^\circ\), then \(\vec{F_2}\) is in the opposite direction of \(\vec{F_1}\):
\[ \vec{F_2} = -b \hat{i} \]

### Conclusion
Thus, the vector \(\vec{F_2}\) could either be in the same direction as \(\vec{F_1}\) or in the opposite direction. Therefore, \(\vec{F_2}\) could be represented as:
\[ \vec{F_2} = b \hat{i} \quad \text{or} \quad \vec{F_2} = -b \hat{i} \]
where \(b\) is any scalar.

### Final Answer
The vector \(\vec{F_2}\) could be any vector along the X-axis, either in the positive or negative direction.

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