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

Answer

To find the magnitude of the vector product of two vectors \(\vec{A}\) and \(\vec{B}\), we can use the formula for the vector (cross) product:

\[ |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin \theta \]

where:
- \(|\vec{A}|\) is the magnitude of vector \(\vec{A}\),
- \(|\vec{B}|\) is the magnitude of vector \(\vec{B}\),
- \(\theta\) is the angle between the two vectors.

### Step 1: Analyze the formula
The sine function, \(\sin \theta\), varies depending on the angle \(\theta\):
- When \(\theta = 0^\circ\), \(\sin 0 = 0\), thus \(|\vec{A} \times \vec{B}| = 0\).
- When \(\theta = 90^\circ\), \(\sin 90 = 1\), thus \(|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}|\).
- For angles between \(0^\circ\) and \(90^\circ\), \(\sin \theta\) will take values between \(0\) and \(1\), meaning \(|\vec{A} \times \vec{B}|\) will take values between \(0\) and \(|\vec{A}| |\vec{B}|\).

### Step 2: Determine the range of \(|\vec{A} \times \vec{B}|\)
From the analysis:
- The minimum value of \(|\vec{A} \times \vec{B}|\) is \(0\) (when \(\theta = 0^\circ\)).
- The maximum value of \(|\vec{A} \times \vec{B}|\) is \(|\vec{A}| |\vec{B}|\) (when \(\theta = 90^\circ\)).
- Therefore, the magnitude of the vector product can be any value in the interval \([0, |\vec{A}| |\vec{B}|]\).

### Step 3: Evaluate the options
Now, we can evaluate the given options:
1. Greater than \(|\vec{A}| |\vec{B}|\) - **Incorrect** (as the maximum is \(|\vec{A}| |\vec{B}|\)).
2. Equal to \(|\vec{A}| |\vec{B}|\) - **Correct** (when \(\theta = 90^\circ\)).
3. Less than \(|\vec{A}| |\vec{B}|\) - **Correct** (for angles \(0 < \theta < 90\)).
4. Equal to \(0\) - **Correct** (when \(\theta = 0^\circ\)).

### Conclusion
The correct answers are:
- Equal to \(|\vec{A}| |\vec{B}|\)
- Less than \(|\vec{A}| |\vec{B}|\)
- Equal to \(0\)