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Math Help - Quick vector question

  1. #1
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    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
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  2. #2
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    Quote Originally Posted by Olym View Post
    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.
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