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

Options

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

Explanation

To solve the problem of determining 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 defined as:
\[ \vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin \theta \, \hat{n} \]
where \( \theta \) is the angle between the two vectors, and \( \hat{n} \) is the unit vector perpendicular to the plane containing \( \vec{A} \) and \( \vec{B} \).

### Step 2: Calculate the Magnitude of the Vector Product
The magnitude of the vector product is given by:
\[ |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin \theta \]

### Step 3: Analyze the Range of \( \sin \theta \)
The sine function varies between -1 and 1. Therefore, the value of \( |\sin \theta| \) can range from 0 to 1. This leads us to the following conclusions:
- If \( \theta = 0 \) or \( \theta = \pi \) (the vectors are parallel), then \( \sin \theta = 0 \) and hence \( |\vec{A} \times \vec{B}| = 0 \).
- If \( \theta = 90^\circ \) (the vectors are perpendicular), then \( \sin \theta = 1 \) and \( |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \).

### Step 4: Establish Relationships
From the above analysis, we can establish that:
\[ 0 \leq |\vec{A} \times \vec{B}| \leq |\vec{A}| |\vec{B}| \]
This means the magnitude of the vector product is always less than or equal to the product of the magnitudes of the two vectors.

### Step 5: Conclusion
Based on the analysis, we can conclude:
- The magnitude of the vector product \( |\vec{A} \times \vec{B}| \) can be equal to 0 (when vectors are parallel).
- The magnitude can be less than \( |\vec{A}| |\vec{B}| \) (when the angle is not 90 degrees).
- The magnitude can be equal to \( |\vec{A}| |\vec{B}| \) only when the vectors are perpendicular.

Thus, the possible answers are:
- Less than \( |\vec{A}| |\vec{B}| \)
- Equal to \( |\vec{A}| |\vec{B}| \) when \( \theta = 90^\circ \)
- Equal to 0 when \( \theta = 0 \) or \( \theta = \pi \)

### Final Answer
The magnitude of the vector product of two vectors \( \vec{A} \) and \( \vec{B} \) can be:
- Less than or equal to \( |\vec{A}| |\vec{B}| \)
- Equal to 0