There exist two different complex numbers $c_1$ and $c_2$, that together with $2+2i, 5+i$ form the vertices of two equilateral triangles. Find the product $c_1c_2$.
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You should say what you tried, or at least what methods you are expected to use. – Conifold Sep 06 '14 at 23:44
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How can 4 points form the vertices of an equilateral triangle? – dfg Sep 06 '14 at 23:47
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@dfg no, the two given points and c1 OR c2 can make the triangle. The two triangles share the side with the two given points, but they face in different directions. – Asimov Sep 06 '14 at 23:49
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@dfg, yeah, i can understand your confusion – Asimov Sep 06 '14 at 23:51
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In the complex plane, this is just a simple geometry problem
So, two points of the triangle are (2,2) and (5,1).
The link below shows how to determine a third point given 2 points and the side lengths Hopefully with it you should be able to figure out the rest. Comment if you get lost so I can advise.
Determine third point of triangle when two points and all sides are known?
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Actually, you are right, that isnt a great example. So, do you know how a dot product works? Well, we know all the side lengths, and that they are supposed to be $\sqrt{10}$ long, we you need to pick coordinates so that the dot product is | x|| y| cosθ where x and y are the vectors (one from the given point to the other, and one to the unknown point) cos 60 degrees is 1/2 and $\sqrt{10}*\sqrt{10}=10$, so the dot product has to be 5 – Asimov Sep 07 '14 at 00:07
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and the dot product can also be computed as $(x_1x_2)+(y_1y_2)$ where $(x_1,y_1)$ is one vector, and $(x_2,y_2)$ is the other. – Asimov Sep 07 '14 at 00:09
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So, what you need to do is find coordinates that make it equal 5,and the two solutions should be your two points (yes, its complicated, but it works) – Asimov Sep 07 '14 at 00:10
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So i need to coordinates that make (x1x2) + (y1y2) = 5? Do i just guess and check? – Aditya More Sep 07 '14 at 00:16
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well, one of those (x1,y1) is the vector from one point to the other, or it is <3,-1>. (from point 1 to point 2) the other <x2,y2> is the vector, from point 1, to the corner you are looking for, or one of the unknowns. There should be 2 solutions – Asimov Sep 07 '14 at 00:51
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The use of vectors in geometry is a complete topic. I cant just teach you how to use them in this comment string – Asimov Sep 07 '14 at 02:50
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Basically a vector is not a point, but a direction and distance, and they can represent motion, or sides of objects, or energy, or many things – Asimov Sep 07 '14 at 02:51