Review the graphs of complex numbers z and w.

On a coordinate plane, the y-axis is labeled imaginary and the x-axis is labeled real. Point z is at (3, 1) and point w is at (negative 5, 5). A line goes from (0, 0) to point z, and from (0, 0) to point w.

What is the product of z and w?

–20 + 10i

–15 + 5i

8 – 4i

20 – 10i

To find the product of two complex numbers \( z \) and \( w \), represented as points on the complex plane, we can use the following approach:

1. **Write the points as complex numbers:**  
   - Point \( z = (3, 1) \) represents the complex number \( z = 3 + i \).
   - Point \( w = (-5, 5) \) represents the complex number \( w = -5 + 5i \).

2. **Calculate the product \( z \times w \):**  
   \[
   (3 + i) \times (-5 + 5i)
   \]
   Using the distributive property:
   \[
   = 3 \times (-5) + 3 \times (5i) + i \times (-5) + i \times (5i)
   \]
   \[
   = -15 + 15i - 5i + 5i^2
   \]
   Recall that \( i^2 = -1 \), so:
   \[
   = -15 + 10i + 5(-1)
   \]
   \[
   = -15 + 10i - 5
   \]
   \[
   = -20 + 10i
   \]

Thus, the product of \( z \) and \( w \) is:

\[
\boxed{-20 + 10i}
\]

Would you like more details, or have any questions?

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Here are five related questions to expand on this topic:

1. How would the product change if we multiplied by the conjugates of \( z \) and \( w \)?
2. What is the geometric interpretation of multiplying two complex numbers?
3. How does the angle of the product compare to the angles of \( z \) and \( w \)?
4. Can you verify the product using polar coordinates instead of rectangular form?
5. How can you determine the modulus of the product based on the moduli of \( z \) and \( w \)?

**Tip:** When multiplying complex numbers, remember to apply \( i^2 = -1 \) in calculations to simplify terms.

Explore how to find the product of two complex numbers, z = 3 + i and w = -5 + 5i, by using the distributive property. Learn to simplify the multiplication, identify real and imaginary components, and interpret the result in the form -20 + 10i. Suitable for Grades 10-12.

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