Question

the relationship between voltage, \\(e\\), current, \\(i\\), and resistance, \\(z\\), is given by the equation \\(e = iz\\). if a circuit has a current \\(i = 3 - 2i\\) and a resistance \\(z = 2 - i\\), what is the voltage of the circuit?

\\(\bigcirc\\) \\(4 - 7i\\)
\\(\bigcirc\\) \\(8 - 7i\\)
\\(\bigcirc\\) \\(8 + 7i\\)
\\(\bigcirc\\) \\(8 - i\\)

Explanation:

⚡ Using what you learned: Introduction to Complex Numbers

Step 1: Identify the given values

The relationship is given by:

$$ E = IZ $$

From the problem statement:

• Current, \( I = 3 - 2i \)
• Resistance, \( Z = 2 - i \)

Step 2: Multiply the complex numbers

Substitute the values into the equation:

$$ E = (3 - 2i)(2 - i) $$

Expand using the FOIL method:

$$ E = 3(2) + 3(-i) + (-2i)(2) + (-2i)(-i) $$

$$ E = 6 - 3i - 4i + 2i^2 $$

Step 3: Simplify using \( i^2 = -1 \)

Substitute \( i^2 = -1 \) into the expression:

$$ E = 6 - 7i + 2(-1) $$

$$ E = 6 - 7i - 2 $$

$$ E = 4 - 7i $$

Answer:

\( 4 - 7i \)