1. **State the problem:** We have two unit vectors $\mathbf{A}$ and $\mathbf{B}$ such that their sum is $\mathbf{A} + \mathbf{B} = 1$. We want to find the dot product $\mathbf{A} \cdot \mathbf{B}$. 2. **Recall the properties:** Since $\mathbf{A}$ and $\mathbf{B}$ are unit vectors, their magnitudes are $|\mathbf{A}| = 1$ and $|\mathbf{B}| = 1$. 3. **Use the formula for the magnitude of a sum:** $$ |\mathbf{A} + \mathbf{B}|^2 = |\mathbf{A}|^2 + |\mathbf{B}|^2 + 2(\mathbf{A} \cdot \mathbf{B}) $$ 4. **Substitute the known values:** $$ 1^2 = 1^2 + 1^2 + 2(\mathbf{A} \cdot \mathbf{B}) $$ 5. **Simplify:** $$ 1 = 1 + 1 + 2(\mathbf{A} \cdot \mathbf{B}) $$ 6. **Rearrange to solve for $\mathbf{A} \cdot \mathbf{B}$:** $$ 1 = 2 + 2(\mathbf{A} \cdot \mathbf{B}) $$ $$ 1 - 2 = 2(\mathbf{A} \cdot \mathbf{B}) $$ $$ -1 = 2(\mathbf{A} \cdot \mathbf{B}) $$ $$ \mathbf{A} \cdot \mathbf{B} = \frac{-1}{2} $$ 7. **Final answer:** $$ \mathbf{A} \cdot \mathbf{B} = -0.5 $$ This means the dot product of the two unit vectors is $-0.5$.