To solve these problems using phasors, we first need to convert the given time-domain functions into their phasor equivalents. ### Step 1: Convert Time-Domain Functions to Phasors 1. **\( v_1(t) = 10 \cos(100t + 30^\circ) \)** - Phasor: \( \mathbf{V}_1 = 10 \angle 30^\circ \) 2. **\( v_2(t) = 3 \cos(100t + 60^\circ) \)** - Phasor: \( \mathbf{V}_2 = 3 \angle 60^\circ \) 3. **\( i_1(t) = 2 \cos(100t - 10^\circ) \)** - Phasor: \( \mathbf{I}_1 = 2 \angle -10^\circ \) ### Step 2: Perform the Required Operations 1. **\( \mathbf{V}_1 + \mathbf{V}_2 \)** - Convert to rectangular form: - \( \mathbf{V}_1 = 10 \angle 30^\circ = 10 (\cos 30^\circ + j \sin 30^\circ) = 10 (\frac{\sqrt{3}}{2} + j \frac{1}{2}) = 5\sqrt{3} + j5 \) - \( \mathbf{V}_2 = 3 \angle 60^\circ = 3 (\cos 60^\circ + j \sin 60^\circ) = 3 (\frac{1}{2} + j \frac{\sqrt{3}}{2}) = 1.5 + j1.5\sqrt{3} \) - Addition: \[ \mathbf{V}_1 + \mathbf{V}_2 = (5\sqrt{3} + 1.5) + j(5 + 1.5\sqrt{3}) \] 2. **\( \mathbf{V}_1 - \mathbf{V}_2 \)** - Subtraction: \[ \mathbf{V}_1 - \mathbf{V}_2 = (5\sqrt{3} - 1.5) + j(5 - 1.5\sqrt{3}) \] 3. **\( \mathbf{V}_1 \cdot \mathbf{I}_1^* \)** - Conjugate of \( \mathbf{I}_1 \): \( \mathbf{I}_1^* = 2 \angle 10^\circ \) - Multiplication: \[ \mathbf{V}_1 \cdot \mathbf{I}_1^* = 10 \angle 30^\circ \cdot 2 \angle 10^\circ = 20 \angle (30^\circ + 10^\circ) = 20 \angle 40^\circ \] 4. **\( \mathbf{V}_1 / \mathbf{I}_1 \)** - Division: \[ \frac{\mathbf{V}_1}{\mathbf{I}_1} = \frac{10 \angle 30^\circ}{2 \angle -10^\circ} = 5 \angle (30^\circ + 10^\circ) = 5 \angle 40^\circ \] 5. **\( V_1 \cdot I_1 \)** - Direct multiplication: \[ V_1 \cdot I_1 = 10 \times 2 \angle (30^\circ - 10^\circ) = 20 \angle 20^\circ \] 6. **\( V_1 / I_1 \)** - Division: \[ \frac{V_1}{I_1} = \frac{10}{2} \angle (30^\circ + 10^\circ) = 5 \angle 40^\circ \] ### Final Answers 1. \( \mathbf{V}_1 + \mathbf{V}_2 = (5\sqrt{3} + 1.5) + j(5 + 1.5\sqrt{3}) \) 2. \( \mathbf{V}_1 - \mathbf{V}_2 = (5\sqrt{3} - 1.5) + j(5 - 1.5\sqrt{3}) \) 3. \( \mathbf{V}_1 \cdot \mathbf{I}_1^* = 20 \angle 40^\circ \) 4. \( \mathbf{V}_1 / \mathbf{I}_1 = 5 \angle 40^\circ \) 5. \( V_1 \cdot I_1 = 20 \angle 20^\circ \) 6. \( V_1 / I_1 = 5 \angle 40^\circ \)