The magnitude of the vectors product of two vectors `|vecA| and |vecB|` may be

Options

A. greater than AB
B. equal to AB
C. less than AB
D. equal to zero

Explanation

To solve the question regarding the magnitude of the vector product of two vectors \(|\vec{A}|\) and \(|\vec{B}|\), we can follow these steps:

### Step 1: Understand the Vector Product
The vector product (or cross product) of two vectors \(\vec{A}\) and \(\vec{B}\) is given by the formula:
\[ |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin \theta \]
where \(\theta\) is the angle between the two vectors.

### Step 2: Analyze the Range of \(\sin \theta\)
The sine function, \(\sin \theta\), varies between 0 and 1 for angles \(\theta\) ranging from 0° to 180°. Therefore, we can deduce:
\[ 0 \leq \sin \theta \leq 1 \]

### Step 3: Determine the Magnitude of the Vector Product
From the formula for the magnitude of the vector product, we can conclude:
\[ 0 \leq |\vec{A} \times \vec{B}| \leq |\vec{A}| |\vec{B}| \]
This means that the magnitude of the vector product is always less than or equal to the product of the magnitudes of the two vectors.

### Step 4: Identify Specific Cases
1. **Equal to \( |\vec{A}| |\vec{B}| \)**: This occurs when \(\sin \theta = 1\), which happens when \(\theta = 90^\circ\) (the vectors are perpendicular).
2. **Less than \( |\vec{A}| |\vec{B}| \)**: This occurs when \(0 < \theta < 90^\circ\) or \(90^\circ < \theta < 180^\circ\), where \(\sin \theta < 1\).
3. **Equal to 0**: This occurs when \(\theta = 0^\circ\) or \(\theta = 180^\circ\) (the vectors are parallel).

### Conclusion
The magnitude of the vector product of two vectors \(|\vec{A}|\) and \(|\vec{B}|\) can be:
- **Equal to \( |\vec{A}| |\vec{B}| \)** (when \(\theta = 90^\circ\))
- **Less than \( |\vec{A}| |\vec{B}| \)** (when \(0 < \theta < 90^\circ\) or \(90^\circ < \theta < 180^\circ\))
- **Equal to 0** (when \(\theta = 0^\circ\) or \(\theta = 180^\circ\))

Thus, the answer to the question is that the magnitude of the vector product can be equal to \( |\vec{A}| |\vec{B}|\), less than \( |\vec{A}| |\vec{B}|\), or equal to 0.