4. The following information is given in an electrical circuit: \( z_{1}=-3+2 i, z_{2}=-2+5 i \) and \( z_{3}=-2-3 i \) Determine the value of \( z_{\mathrm{p}} \), in rectangular form, if: \( z_{\mathrm{p}}=\frac{z_{1} \times z_{2}}{z_{1}+z_{2}}+z_{3} \)

To determine the value of \( z_{\mathrm{p}} \), let's go through the calculations step by step! First, calculate \( z_{1} \times z_{2} \): \[ z_{1} \times z_{2} = (-3 + 2i)(-2 + 5i) \] Using the distributive property (FOIL): \[ = (-3 \times -2) + (-3 \times 5i) + (2i \times -2) + (2i \times 5i) \] \[ = 6 - 15i - 4i + 10(-1) \] \[ = 6 - 15i - 4i - 10 = -4 - 19i \] Next, calculate \( z_{1} + z_{2} \): \[ z_{1} + z_{2} = (-3 + 2i) + (-2 + 5i) = -3 - 2 + (2 + 5)i = -5 + 7i \] Now, substitute \( z_{1} \times z_{2} \) and \( z_{1} + z_{2} \) into the formula for \( z_{\mathrm{p}} \): \[ z_{\mathrm{p}} = \frac{-4 - 19i}{-5 + 7i} + z_{3} \] To perform the division \( \frac{-4 - 19i}{-5 + 7i} \), multiply the numerator and denominator by the conjugate of the denominator: \[ = \frac{(-4 - 19i)(-5 - 7i)}{(-5 + 7i)(-5 - 7i)} \] Calculating the denominator: \[ (-5 + 7i)(-5 - 7i) = 25 + 49 = 74 \] Now for the numerator: \[ (-4 - 19i)(-5 - 7i) = 20 + 28i + 95i + 133 = 153 + 123i \] So we have: \[ \frac{-4 - 19i}{-5 + 7i} = \frac{153 + 123i}{74} = \frac{153}{74} + \frac{123}{74}i \] Next, we can simplify \( z_{\mathrm{p}} \) by adding \( z_{3} \): \[ z_{3} = -2 - 3i \] Now combining: \[ z_{\mathrm{p}} = \left(\frac{153}{74} - 2\right) + \left(\frac{123}{74} - 3\right)i \] Calculating the real part: \[ \frac{153}{74} - \frac{148}{74} = \frac{5}{74} \] Calculating the imaginary part: \[ \frac{123}{74} - \frac{222}{74} = \frac{-99}{74} \] Thus, the value of \( z_{\mathrm{p}} \) in rectangular form is: \[ z_{\mathrm{p}} = \frac{5}{74} - \frac{99}{74}i \]