The magnitude of the vector product of two vectors and may be : `overset(rarr)A` and `overset(rarr)B`

Options

A. (a)greater than AB
B. (b)equal to AB
C. (c)less than AB
D. (d)equal to zero

Explanation

To solve the problem regarding the magnitude of the vector product of two vectors \( \vec{A} \) and \( \vec{B} \), we need to understand the properties of vector products and how they relate to the magnitudes of the vectors involved.

### Step-by-Step Solution:

1. **Understanding 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.

2. **Magnitude of Vectors**: The magnitudes \( |\vec{A}| \) and \( |\vec{B}| \) are simply the lengths of the vectors \( \vec{A} \) and \( \vec{B} \).

3. **Range of Sine Function**: The sine function \( \sin \theta \) varies between -1 and 1, but since we are interested in the magnitude, we consider only the positive values:
\[ 0 \leq \sin \theta \leq 1 \]

4. **Maximum and Minimum Values**:
- The maximum value of \( |\vec{A} \times \vec{B}| \) occurs when \( \sin \theta = 1 \) (i.e., when \( \theta = 90^\circ \)):
\[ |\vec{A} \times \vec{B}|_{\text{max}} = |\vec{A}| |\vec{B}| \]
- The minimum value occurs when \( \sin \theta = 0 \) (i.e., when \( \theta = 0^\circ \) or \( 180^\circ \)):
\[ |\vec{A} \times \vec{B}|_{\text{min}} = 0 \]

5. **Conclusion**: Therefore, the magnitude of the vector product \( |\vec{A} \times \vec{B}| \) can take any value in the range:
\[ 0 \leq |\vec{A} \times \vec{B}| \leq |\vec{A}| |\vec{B}| \]

6. **Options Analysis**: Based on the above analysis, we can conclude that:
- The magnitude of the vector product can be equal to zero (when the vectors are parallel).
- The magnitude can be less than or equal to the product of the magnitudes of the vectors.

### Final Answer:
The correct options regarding the magnitude of the vector product of \( \vec{A} \) and \( \vec{B} \) are:
- It can be equal to \( 0 \).
- It can be less than or equal to \( |\vec{A}| |\vec{B}| \).