If `|a+b|= |a-b|`, then which one of the following is correct?

Options

A. `|a|=|b|`
B. a is parallel to b
C. a is perpendicular to b
D. `vec(a)` is a unit vector

Answer

To solve the problem, we start with the equation given:

**Given:**
\[ |a + b| = |a - b| \]

### Step 1: Square both sides
To eliminate the absolute values, we square both sides of the equation:
\[ (|a + b|)^2 = (|a - b|)^2 \]

### Step 2: Expand both sides
Using the property of squares, we expand both sides:
\[ (a + b) \cdot (a + b) = (a - b) \cdot (a - b) \]
This gives us:
\[ |a|^2 + 2(a \cdot b) + |b|^2 = |a|^2 - 2(a \cdot b) + |b|^2 \]

### Step 3: Simplify the equation
Next, we simplify the equation by canceling out the common terms:
\[ |a|^2 + |b|^2 + 2(a \cdot b) = |a|^2 + |b|^2 - 2(a \cdot b) \]
Subtracting \(|a|^2 + |b|^2\) from both sides, we get:
\[ 2(a \cdot b) = -2(a \cdot b) \]

### Step 4: Rearranging the equation
Now, we can rearrange the equation:
\[ 2(a \cdot b) + 2(a \cdot b) = 0 \]
This simplifies to:
\[ 4(a \cdot b) = 0 \]

### Step 5: Conclusion
From the equation \(4(a \cdot b) = 0\), we conclude:
\[ a \cdot b = 0 \]
This implies that vectors \(a\) and \(b\) are perpendicular to each other.

### Final Answer
Thus, the correct option is that \(a\) and \(b\) are perpendicular.

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