Complex Arithmetic/Examples/3(1+i) + 2(4-3i) - (2+5i)/Proof 2

Proof
We have:

These can be depicted in the complex plane as follows:


 * Complex-Addition-(3(1+i))+(2(4-3i))-(2+5i)--1.png

To find the required sum, proceed as in the following diagram:


 * Complex-Addition-(3(1+i))+(2(4-3i))-(2+5i)--2.png

Construct $8 - 6 i$ with its initial point placed at the terminal point of $3 + 3 i$.

Construct $-2 - 5 i$ with its initial point placed at the terminal point of this instance of $8 - 6 i$.

The required resultant $3 \paren {1 + i} + 2 \paren {4 - 3 i} - \paren {2 + 5 i}$ is therefore represented by the terminal point of $-2 - 5 i$.