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# Complex Numbers with a Property (TOMATO Subjective 88)

Problem: A pair of complex numbers $$z_1, z_2$$ is said to have the property $$P$$ if for every complex number $$z$$ we find real numbers $$r$$ and $$s$$ such that $$z=rz_1 + sz_2$$.Show that a pair of complex numbers has property $$P$$ if and only if the points $$z_1,z_2$$ and $$0$$ on the complex plane are not collinear.

Solution: Let the complex numbers $$z_1,z_2,0$$ be collinear, and the line joining them make an angle $$\theta$$ with the x-axis. This means that:

$$arg(z_1) =arg(z_2) = \theta$$

$$=> z_1 = |z_1| (cos\,\theta + i sin\, \theta)$$

October 16, 2016