If `(|veca+vecb|)/(|veca-vecb|)=1`, then angle between `bar(a)" and "bar(b)` is :-

Options

A. `0^(@)`
B. `45^(@)`
C. `90^(@)`
D. `60^(@)`

Answer

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

\[ \frac{|\vec{a} + \vec{b}|}{|\vec{a} - \vec{b}|} = 1 \]

### Step 1: Set up the equation
Since the ratio is equal to 1, we can write:

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

### Step 2: Square both sides
To eliminate the magnitudes, we square both sides:

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

### Step 3: Expand both sides
Using the formula for the magnitude of a vector, we expand both sides:

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

This gives us:

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

### Step 4: Simplify the equation
Now, we can simplify the equation by canceling out the common terms on both sides:

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

### Step 5: Combine like terms
Adding \(2\vec{a} \cdot \vec{b}\) to both sides gives:

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

### Step 6: Solve for the angle
Since \(4\vec{a} \cdot \vec{b} = 0\), we can conclude that:

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

The dot product of two vectors is zero when the angle between them is \(90^\circ\). Thus, we find:

\[ \theta = 90^\circ \]

### Final Answer
The angle between \(\vec{a}\) and \(\vec{b}\) is \(90^\circ\).

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