The complex number \( w \) is given by \( w=-\frac{1}{2}+\frac{\sqrt{3} i}{2} \) State the modulus of wz given that \( z=1+i \). Select one: a. \( \sqrt{4} \) b. \( \frac{\sqrt{2}}{2} \) c. \( \sqrt{2} \) d. \( \frac{1}{2} \)

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