hich quadrant is  z1+z2−z3𝑧1+𝑧2−𝑧3ifz1=2ej(π/2)𝑧1=2𝑒𝑗(𝜋/2)z2=2+j3𝑧2=2+𝑗3z3=2–√(cosπ4+jsinπ4)𝑧3=2(cos⁡𝜋4+𝑗sin⁡𝜋4)? Question 1Select one:13None of the above42

1. Break Down the Problem

We need to find the quadrant of the expression $z_1 + z_2 - z_3$, given the following complex numbers:

• $z_1 = 2e^{j(\pi/2)}$
• $z_2 = 2 + j3$
• $z_3 = 2\left(\cos\frac{\pi}{4} + j\sin\frac{\pi}{4}\right)$

2. Relevant Concepts

1. Convert all complex numbers into rectangular form.
2. Calculate $z_1 + z_2 - z_3$.
3. Identify the quadrant of the resulting complex number based on its real and imaginary parts.

3. Analysis and Detail

Step 1: Convert $z_1$ and $z_3$ into rectangular form.

• $z_1 = 2e^{j(\pi/2)} = 2(\cos(\pi/2) + j\sin(\pi/2))$ $z_1 = 2(0 + j \cdot 1) = 2j$

• $z_3 = 2\left(\cos\frac{\pi}{4} + j\sin\frac{\pi}{4}\right)$ $z_3 = 2\left(\frac{\sqrt{2}}{2} + j\frac{\sqrt{2}}{2}\right) = 2\cdot\frac{\sqrt{2}}{2} + j 2\cdot\frac{\sqrt{2}}{2} = \sqrt{2} + j\sqrt{2}$

Step 2: Calculate $z_1 + z_2 - z_3$.

• Now we write $z_2$ in rectangular form: $z_2 = 2 + j3$

• Combine the values: $z_1 + z_2 - z_3 = (2j) + (2 + j3) - (\sqrt{2} + j\sqrt{2})$

Calculating the real and imaginary parts: $\text{Real part: } 2 - \sqrt{2}, \quad \text{Imaginary part: } 2 + 3 - \sqrt{2} = 5 - \sqrt{2}$

The expression becomes: $z_1 + z_2 - z_3 = (2 - \sqrt{2}) + j(5 - \sqrt{2})$

4. Verify and Summarize

1. The real part: $2 - \sqrt{2}$

   • Approximately $2 - 1.414 = 0.586$ (positive)

2. The imaginary part: $5 - \sqrt{2}$

   • Approximately $5 - 1.414 = 3.586$ (positive)

Final Answer

Since both the real and imaginary parts are positive, the complex number $z_1 + z_2 - z_3$ lies in the first quadrant.

Select one: 1 (First Quadrant)