If `u = a-b, v =a + b and |a| = |b| = 2,` then `|u xx v| ` is

Options

A. `2sqrt(16-(a.b)^(2))`
B. `2sqrt(4-(a.b)^(2))`
C. `sqrt(16-(a.b)^(2))`
D. `sqrt(4-(a.b)^(2))`

Answer

To solve the problem, we need to find the magnitude of the cross product of the vectors \( u \) and \( v \), where \( u = a - b \) and \( v = a + b \), and the magnitudes of vectors \( a \) and \( b \) are given as \( |a| = |b| = 2 \).

### Step-by-Step Solution:

1. **Define the vectors**:
- Let \( u = a - b \)
- Let \( v = a + b \)

2. **Calculate the cross product \( u \times v \)**:
\[ u \times v = (a - b) \times (a + b) \]
Using the distributive property of the cross product:
\[ u \times v = a \times a + a \times b - b \times a - b \times b \]

3. **Simplify the cross product**:
- The cross product of any vector with itself is zero:
\[ a \times a = 0 \quad \text{and} \quad b \times b = 0 \]
- Therefore, we have:
\[ u \times v = 0 + a \times b - b \times a + 0 = a \times b - b \times a \]
- Since \( b \times a = - (a \times b) \), we can write:
\[ u \times v = a \times b + a \times b = 2(a \times b) \]

4. **Find the magnitude of \( u \times v \)**:
\[ |u \times v| = |2(a \times b)| = 2 |a \times b| \]

5. **Calculate the magnitude of \( a \times b \)**:
The magnitude of the cross product can be calculated using the formula:
\[ |a \times b| = |a| |b| \sin \theta \]
where \( \theta \) is the angle between vectors \( a \) and \( b \). Given \( |a| = |b| = 2 \):
\[ |a \times b| = 2 \cdot 2 \cdot \sin \theta = 4 \sin \theta \]

6. **Substitute back into the magnitude**:
\[ |u \times v| = 2 |a \times b| = 2(4 \sin \theta) = 8 \sin \theta \]

7. **Final expression**:
Thus, the final expression for the magnitude of \( u \times v \) is:
\[ |u \times v| = 8 \sin \theta \]

### Conclusion:
The magnitude of \( |u \times v| \) is \( 8 \sin \theta \).