Compute \( \mathbf{u} \cdot \mathbf{v} \) if \( \mathbf{u} \) and \( \mathbf{v} \) are unit vectors and the angle between them is \( \frac{\pi}{4} \). \( \mathbf{u} \cdot \mathbf{v}=\square \) (Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)

To compute the dot product \( \mathbf{u} \cdot \mathbf{v} \) of two unit vectors \( \mathbf{u} \) and \( \mathbf{v} \) with an angle \( \theta \) between them, we use the formula: \[ \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos(\theta). \] Since both \( \mathbf{u} \) and \( \mathbf{v} \) are unit vectors, their magnitudes are: \[ |\mathbf{u}| = 1, \quad |\mathbf{v}| = 1. \] Thus, the dot product simplifies to: \[ \mathbf{u} \cdot \mathbf{v} = 1 \cdot 1 \cdot \cos\left(\frac{\pi}{4}\right). \] Now, we need to evaluate \( \cos\left(\frac{\pi}{4}\right) \). We know that: \[ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}. \] Therefore, we have: \[ \mathbf{u} \cdot \mathbf{v} = 1 \cdot 1 \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}. \] Finally, the answer is: \[ \mathbf{u} \cdot \mathbf{v} = \frac{\sqrt{2}}{2}. \]