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

Options

A. `vec(a)` is parallel to `vec(b)`
B. `vec(a)` is perpendicular to `vec(b)`
C. `vec(a)=vec(b)`
D. Both `vec(a) and vec(b)` are unit vectors

Explanation

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 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 simplifies 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: Cancel common terms

Now, we can cancel \(\vec{a} \cdot \vec{a}\) and \(\vec{b} \cdot \vec{b}\) from both sides:

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

### Step 4: Rearrange the equation

Bringing all terms involving \(\vec{a} \cdot \vec{b}\) to one side gives:

\[ 2\vec{a} \cdot \vec{b} + 2\vec{a} \cdot \vec{b} = 0 \]

This simplifies to:

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

### Step 5: Conclusion

From this, we conclude that:

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

This implies that vectors \(\vec{a}\) and \(\vec{b}\) are perpendicular to each other.

### Final Answer

Thus, the correct option is that \(\vec{a}\) is perpendicular to \(\vec{b}\).

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