To determine which graph represents the sum [tex]\( w_1 + w_2 \)[/tex], let's go through the detailed calculation step-by-step.
Given two complex numbers:
[tex]\[ w_1 = 4 + 2i \][/tex]
[tex]\[ w_2 = -1 - 3i \][/tex]
Step 1: Add the Real Parts
First, we add the real parts of [tex]\( w_1 \)[/tex] and [tex]\( w_2 \)[/tex]:
[tex]\[ \text{Real part of } w_1 = 4 \][/tex]
[tex]\[ \text{Real part of } w_2 = -1 \][/tex]
[tex]\[ \text{Sum of real parts} = 4 + (-1) = 3 \][/tex]
Step 2: Add the Imaginary Parts
Next, we add the imaginary parts of [tex]\( w_1 \)[/tex] and [tex]\( w_2 \)[/tex]:
[tex]\[ \text{Imaginary part of } w_1 = 2 \][/tex]
[tex]\[ \text{Imaginary part of } w_2 = -3 \][/tex]
[tex]\[ \text{Sum of imaginary parts} = 2 + (-3) = -1 \][/tex]
Step 3: Combine the Results
We now combine the sums of the real and imaginary parts to get the resultant complex number.
[tex]\[ w_1 + w_2 = 3 - 1i \][/tex]
So, the sum of the complex numbers [tex]\( w_1 \)[/tex] and [tex]\( w_2 \)[/tex] is:
[tex]\[ w_1 + w_2 = 3 - 1i \][/tex]
Graph Representation:
A complex number [tex]\( 3 - 1i \)[/tex] can be represented graphically on the complex plane where the horizontal axis is the real part and the vertical axis is the imaginary part. In this case, the point corresponding to [tex]\( 3 - 1i \)[/tex] is plotted at:
[tex]\[ (3, -1) \][/tex]
Therefore, the graph that represents the sum [tex]\( w_1 + w_2 \)[/tex] is the one with the point located at coordinates (3, -1) on the complex plane.
