(i) If `|a + b| = |a -b|`, then a and b are parallel. True or False

Answer

To determine whether the statement "|a + b| = |a - b| implies that vectors a and b are parallel" is true or false, we can follow these steps:

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ |a + b| = |a - b| \]

2. **Square both sides to eliminate the absolute values:**
\[ |a + b|^2 = |a - b|^2 \]

3. **Use the formula for the square of the magnitude of a vector:**
\[ |u|^2 = u \cdot u \]
Therefore, we can write:
\[ (a + b) \cdot (a + b) = (a - b) \cdot (a - b) \]

4. **Expand both sides:**
- Left-hand side:
\[ a \cdot a + 2a \cdot b + b \cdot b = |a|^2 + 2a \cdot b + |b|^2 \]
- Right-hand side:
\[ a \cdot a - 2a \cdot b + b \cdot b = |a|^2 - 2a \cdot b + |b|^2 \]

5. **Set the expanded forms equal to each other:**
\[ |a|^2 + 2a \cdot b + |b|^2 = |a|^2 - 2a \cdot b + |b|^2 \]

6. **Simplify the equation:**
- Subtract \(|a|^2 + |b|^2\) from both sides:
\[ 2a \cdot b = -2a \cdot b \]
- Combine like terms:
\[ 4a \cdot b = 0 \]

7. **Conclude from the dot product:**
\[ a \cdot b = 0 \]
This means that vectors a and b are perpendicular.

### Conclusion:
Since the condition \(a \cdot b = 0\) indicates that a and b are perpendicular, the statement that "|a + b| = |a - b| implies that a and b are parallel" is **False**.