geometric interpretation of complex numbers
Hello, I would very much appreciate your looking through my answer to the following question and letting me know any hick ups it may have...
Question: The points A, B, C, in an Argand diagram represent the complex numbers a, b, c, and a = (1-λ)b + λc. Prove that if λ is real then A lies on BC and divides BC in the ratio λ: 1-λ, but if λ is complex then, in triangle ABC, AB:BC = (modulus of λ) : 1 and angle ABC = argλ.
My attempt to answer: λ=x+jy so a = [(1-x)+j(-y)]b+(x+jy)c. If λ is real then y=0 and a is simply the convex combination of b and c and lies on BC dividing it in the ratio λ: 1-λ (:confused: shouldn't λ lie in the interval [0,1] otherwise a will lie on the extension of BC rather than on BC???) If y different to 0 then (1-λ) and λ are complex multiplying b and c respectively. This means that b - resp. c - is turned through the angle arg(1-λ) - resp. argλ - and stretched by the scale factor "modulus of 1-λ" - resp. :modulus of λ" - to give the vector (1-λ)b - resp. λc . Then so long as b and c are distinct ABC is a triangle and AB/AC = (a-b)/(c-b)=[(1-λ)b+λc]/(c-b)=λ and therefore also the angle ABC = argλ. :confused: my answer to the case where λ is complex sounds too confused to be right??? thanks for any help..