If `(|a+b|)/(|a-b|)=1`, then the angle between `a` and `b` is

Options

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

Explanation

To solve the problem, we need to analyze the given equation:

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

This implies that:

\[ |a+b| = |a-b| \]

### Step 1: Write the expressions for magnitudes

The magnitude of a vector sum and difference can be expressed using the cosine of the angle between the vectors.

The magnitude of \(a + b\) is given by:

\[ |a + b| = \sqrt{|a|^2 + |b|^2 + 2|a||b|\cos\theta} \]

The magnitude of \(a - b\) is given by:

\[ |a - b| = \sqrt{|a|^2 + |b|^2 - 2|a||b|\cos\theta} \]

### Step 2: Set the magnitudes equal

Since we have \( |a + b| = |a - b| \), we can set the two expressions equal to each other:

\[ \sqrt{|a|^2 + |b|^2 + 2|a||b|\cos\theta} = \sqrt{|a|^2 + |b|^2 - 2|a||b|\cos\theta} \]

### Step 3: Square both sides

To eliminate the square roots, we square both sides:

\[ |a|^2 + |b|^2 + 2|a||b|\cos\theta = |a|^2 + |b|^2 - 2|a||b|\cos\theta \]

### Step 4: Simplify the equation

By simplifying the equation, we can cancel out \( |a|^2 + |b|^2 \) from both sides:

\[ 2|a||b|\cos\theta = -2|a||b|\cos\theta \]

### Step 5: Rearranging the equation

Rearranging gives us:

\[ 2|a||b|\cos\theta + 2|a||b|\cos\theta = 0 \]

This simplifies to:

\[ 4|a||b|\cos\theta = 0 \]

### Step 6: Solve for \(\cos\theta\)

Since \( |a| \) and \( |b| \) are non-zero (assuming they are non-zero vectors), we can divide both sides by \( 4|a||b| \):

\[ \cos\theta = 0 \]

### Step 7: Determine the angle

The angle \(\theta\) for which \(\cos\theta = 0\) is:

\[ \theta = 90^\circ \]

### Conclusion

Thus, the angle between vectors \(a\) and \(b\) is \(90^\circ\).