Simplify : (i) `| vec(a).vec(b)|^(2) +| vec(a) xx vec(b)|^(2)` (ii) `| vec(a).vec(b)|^(2)- |vec(a) xx vec(b)|^(2)`

Explanation

To simplify the given expressions, we will use the properties of dot and cross products of vectors. Let's break down the solution step by step for both parts.

### Part (i): Simplify \( |\vec{a} \cdot \vec{b}|^2 + |\vec{a} \times \vec{b}|^2 \)

1. **Recall the definitions**:
- The dot product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]
- The cross product of two vectors \( \vec{a} \) and \( \vec{b} \) is given by:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \]

2. **Calculate \( |\vec{a} \cdot \vec{b}|^2 \)**:
\[ |\vec{a} \cdot \vec{b}|^2 = (|\vec{a}| |\vec{b}| \cos \theta)^2 = |\vec{a}|^2 |\vec{b}|^2 \cos^2 \theta \]

3. **Calculate \( |\vec{a} \times \vec{b}|^2 \)**:
\[ |\vec{a} \times \vec{b}|^2 = (|\vec{a}| |\vec{b}| \sin \theta)^2 = |\vec{a}|^2 |\vec{b}|^2 \sin^2 \theta \]

4. **Combine the results**:
\[ |\vec{a} \cdot \vec{b}|^2 + |\vec{a} \times \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 \cos^2 \theta + |\vec{a}|^2 |\vec{b}|^2 \sin^2 \theta \]

5. **Factor out common terms**:
\[ = |\vec{a}|^2 |\vec{b}|^2 (\cos^2 \theta + \sin^2 \theta) \]

6. **Use the Pythagorean identity**:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]

7. **Final result for part (i)**:
\[ = |\vec{a}|^2 |\vec{b}|^2 \cdot 1 = |\vec{a}|^2 |\vec{b}|^2 \]

### Part (ii): Simplify \( |\vec{a} \cdot \vec{b}|^2 - |\vec{a} \times \vec{b}|^2 \)

1. **Using the results from part (i)**:
- We already have:
\[ |\vec{a} \cdot \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 \cos^2 \theta \]
\[ |\vec{a} \times \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 \sin^2 \theta \]

2. **Combine the results**:
\[ |\vec{a} \cdot \vec{b}|^2 - |\vec{a} \times \vec{b}|^2 = |\vec{a}|^2 |\vec{b}|^2 \cos^2 \theta - |\vec{a}|^2 |\vec{b}|^2 \sin^2 \theta \]

3. **Factor out common terms**:
\[ = |\vec{a}|^2 |\vec{b}|^2 (\cos^2 \theta - \sin^2 \theta) \]

4. **Use the double angle identity**:
\[ \cos^2 \theta - \sin^2 \theta = \cos 2\theta \]

5. **Final result for part (ii)**:
\[ = |\vec{a}|^2 |\vec{b}|^2 \cos 2\theta \]

### Summary of Results
- For part (i): \( |\vec{a}|^2 |\vec{b}|^2 \)
- For part (ii): \( |\vec{a}|^2 |\vec{b}|^2 \cos 2\theta \)