Let's begin by understanding that we are dealing with complex numbers. A complex number is typically represented as [tex]\( z = a + bi \)[/tex], where [tex]\( a \)[/tex] is the real part and [tex]\( b \)[/tex] is the imaginary part.
Given:
[tex]\[ z = -2 + 2i \][/tex]
Now, we need to find [tex]\( -2z \)[/tex].
Step-by-Step Solution:
1. Identify the original complex number and its components:
[tex]\[
z = -2 + 2i
\][/tex]
Here, the real part [tex]\( a \)[/tex] is [tex]\(-2\)[/tex] and the imaginary part [tex]\( b \)[/tex] is [tex]\(2\)[/tex].
2. Plot the point representing [tex]\( z \)[/tex] on the complex plane:
The complex plane has a horizontal axis (real axis) and a vertical axis (imaginary axis).
[tex]\[
z = (-2, 2)
\][/tex]
This means we plot the point [tex]\((-2, 2)\)[/tex] on the complex plane.
3. Calculate [tex]\( -2z \)[/tex]:
[tex]\[
-2z = -2(-2 + 2i) = 4 - 4i
\][/tex]
4. Identify the components of [tex]\( -2z \)[/tex]:
The real part of [tex]\( -2z \)[/tex] is [tex]\( 4 \)[/tex] and the imaginary part is [tex]\( -4 \)[/tex].
5. Plot the point representing [tex]\( -2z \)[/tex] on the complex plane:
[tex]\[
-2z = (4, -4)
\][/tex]
This means we plot the point [tex]\((4, -4)\)[/tex] on the complex plane.
Hence, the two points on the graph are:
- [tex]\( z = (-2, 2) \)[/tex]
- [tex]\( -2z = (4, -4) \)[/tex]
These points represent the complex numbers [tex]\( z \)[/tex] and [tex]\( -2z \)[/tex] on the complex plane.
