The complex number \( \mathbf{w} \) is given by \( w=-\frac{1}{2}+\frac{\sqrt{3} i}{2} \). State the modulus of wz given that \( \mathbf{z = 1}+\mathbf{i} \).

To find the modulus of the product \( wz \), we first need to calculate the moduli of \( w \) and \( z \) individually. For the complex number \( w = -\frac{1}{2} + \frac{\sqrt{3}}{2} i \): \[ |w| = \sqrt{\left(-\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1. \] For the complex number \( z = 1 + i \): \[ |z| = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2}. \] Now, the modulus of the product \( wz \) is given by the product of the moduli: \[ |wz| = |w| \cdot |z| = 1 \cdot \sqrt{2} = \sqrt{2}. \] Thus, the modulus of \( wz \) is \( \sqrt{2} \).