Let’s consider 2 complex numbers z1=x1+iy1 and z2=x2+iy2.
The numbers can be represented geometrically on the complex plane (Argand diagram) as 2 points A(x1,y1) and B(x2,y2).
They can also be represented as 2 vectors, both starting from origin O(0,0) towards A(x1,y1) and B(x2,y2) respectively. The (directed) segment OA represents complex number z1 and OB complex number z2.
When we add the vectors we draw parallel lines to OA and OB, which let’s say intersect at C(x3,y3) and the segment OC (diagonal) represents the sum z3=z1+z2.
The reason for that is because on x-axis we plus the x coordinates of the 2 complex numbers x1 and x2; x3=x1+x2 and same on y-axis, y3=y1+y2. So the complex number z3 (the sum) can be written as: z3=(x1+x2)+i(y1+y2) and is represented by the vector OC.
When we minus them we have 2 cases: z1-z2 or z2-z1.
Let’s say we doing first z1-z2. Subtraction is the opposite operation to addition, therefore we can write: z1-z2=z1+(-z2). If z2=x2+iy2, then -z2=-x2-iy2, which can be represented as a vector opposed to OB, let’s say OD, where D(-x2,-y2). Now we are in the first case, when add OA and OD (draw the parallelogram) and get a new vector let’s say OE (the diagonal), which represents the complex number z1-z2=(x1-x2)+i(y1-y2). If you draw the diagram you can notice that OE is parallel and equal to BA. So we can say that the complex number z1-z2 can be represented by the vector BA (originated at B and pointing towards A, where the arrow is). What is essential to understand is that a vector is a whole class of parallel and equal segments having the same direction.
Now if we doing z2-z1, following the same procedure (plus z2 with the opposite of z1, which is –z1=-x1-iy1), drawing the diagram we find that the vector is AB, same in length with BA (z1-z2), but pointing in the opposite direction (arrow at B).
Hope it helps a bit; don’t want to confuse you even more, not easy to explain without drawing.
Have fun!:wave:
