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

Options

A. `veca = lamda vec b` for some scalar `lamda `
B. `vec a `is parallel to `vec b`
C. `vec a` is perpendicualr to `vecb`
D. `veca = vecb = vec 0 `

Answer

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

\[ |\vec{a} + \vec{b}| = |\vec{a} - \vec{b}| \]

### Step 1: Square both sides
Squaring both sides of the equation to eliminate the absolute values gives us:

\[ |\vec{a} + \vec{b}|^2 = |\vec{a} - \vec{b}|^2 \]

### Step 2: Expand both sides
Using the property of dot products, we can expand both sides:

\[ (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b}) = (\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b}) \]

This expands to:

\[ \vec{a} \cdot \vec{a} + 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = \vec{a} \cdot \vec{a} - 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} \]

### Step 3: Simplify the equation
Now, we can simplify both sides:

\[ \vec{a} \cdot \vec{a} + 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = \vec{a} \cdot \vec{a} - 2 \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} \]

Subtract \(\vec{a} \cdot \vec{a} + \vec{b} \cdot \vec{b}\) from both sides:

\[ 2 \vec{a} \cdot \vec{b} = -2 \vec{a} \cdot \vec{b} \]

### Step 4: Combine like terms
Adding \(2 \vec{a} \cdot \vec{b}\) to both sides gives:

\[ 4 \vec{a} \cdot \vec{b} = 0 \]

### Step 5: Solve for \(\vec{a} \cdot \vec{b}\)
Dividing both sides by 4 results in:

\[ \vec{a} \cdot \vec{b} = 0 \]

### Conclusion
The dot product of \(\vec{a}\) and \(\vec{b}\) being zero indicates that the vectors are perpendicular. Thus, the correct option is:

\[ \text{Option C: } \vec{a} \text{ is perpendicular to } \vec{b} \]

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