If ` |a| = |b| = |a + b| = 1, ` then `|a-b|` is equal to

Options

A. 1
B. `sqrt(2)`
C. `sqrt(3)`
D. none of these

Answer

To solve the problem where \( |a| = |b| = |a + b| = 1 \) and we need to find \( |a - b| \), we can follow these steps:

### Step 1: Use the given magnitudes
We know that:
\[ |a| = 1, \quad |b| = 1, \quad |a + b| = 1 \]

### Step 2: Square the magnitude of \( a + b \)
We can start by squaring the magnitude of \( a + b \):
\[ |a + b|^2 = 1^2 = 1 \]
Using the property of magnitudes, we can expand this:
\[ |a + b|^2 = |a|^2 + |b|^2 + 2(a \cdot b) \]
Substituting the known values:
\[ 1 = 1^2 + 1^2 + 2(a \cdot b) \]
This simplifies to:
\[ 1 = 1 + 1 + 2(a \cdot b) \]
\[ 1 = 2 + 2(a \cdot b) \]
### Step 3: Solve for \( a \cdot b \)
Rearranging gives us:
\[ 2(a \cdot b) = 1 - 2 \]
\[ 2(a \cdot b) = -1 \]
\[ a \cdot b = -\frac{1}{2} \]

### Step 4: Use the identity for \( |a - b|^2 \)
Next, we use the identity for the magnitude of the difference:
\[ |a - b|^2 = |a|^2 + |b|^2 - 2(a \cdot b) \]
Substituting the known values:
\[ |a - b|^2 = 1^2 + 1^2 - 2\left(-\frac{1}{2}\right) \]
This simplifies to:
\[ |a - b|^2 = 1 + 1 + 1 = 3 \]

### Step 5: Take the square root
Finally, we take the square root to find \( |a - b| \):
\[ |a - b| = \sqrt{3} \]

Thus, the final answer is:
\[ |a - b| = \sqrt{3} \]