1. ## Quick vector question

I have the following from the question (in the 2D complex coordinate system):

z3 - z1 = r(z2 - z1), where z1, z2, z3 are complex numbers and r is a real number. I have to show that z1, z2 and z3 lie on the same line. In order to solve the rest of the question, it's recommended that

z3 - z1 = r(z2 - z1) should be rewritten in the form Ax + By + C = 0, which I have no idea how to do.

Any help would be greatly appreciated.

Thanks

2. Originally Posted by Olym
I have the following from the question (in the 2D complex coordinate system):
z3 - z1 = r(z2 - z1), where z1, z2, z3 are complex numbers and r is a real number. I have to show that z1, z2 and z3 lie on the same line. In order to solve the rest of the question, it's recommended that
Let $x_i = \text{Re} (z_i )\;\& \,y_i = \text{Im} \left( {z_i } \right)$.
To show that the three are collinear we show that they define the same slope.
Look at the slopes $\frac{{y_2 - y_1 }}{{x_2 - x_1 }}\;\& \,\frac{{y_3 - y_1 }}{{x_3 - x_1 }}$
But given that $z_3-z¬_1=r(z¬_2-z¬_1)$ we get $\frac{{y_3 - y_1 }}{{x_3 - x_1 }} = \frac{{r(y_2 - y_1 )}}{{r(x_2 - x_1 )}}$.

So the have the same slopes.