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What's the relation between complex and vectors......Complex are usually represented as position vectors and during rotation we only rotate the directed line formed by the two complex.Why do they share some properties when they are entirely different?

  • Welcome to stackexchange. This question is likely to be closed as too vague. If you can edit the question to ask something about particular operations, with examples, maybe we can help. – Ethan Bolker Jan 03 '18 at 14:05
  • Can't u edit this question so as to meet the requirements? –  Jan 03 '18 at 14:07
  • They are equivalent with respect to addition but you cannot multiply two vectors to give another 2D vector. The complex numbers form an algebra, not just a vector space. – Paul Jan 03 '18 at 14:10
  • @GENESECT Sorry, nobody can ask your question for you unless it is already not vague, and then maybe we can help sharpen it. – rschwieb Jan 03 '18 at 14:10
  • How do you "form a directed line by two complex"? – rschwieb Jan 03 '18 at 14:11
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    Of course I can't edit for you since I can't figure out what you are trying to ask. And asking us to ask the question as well as answer it is not a good way to get help. Put in the effort to make your question clear. And please no text message shortcuts like "u". – Ethan Bolker Jan 03 '18 at 14:13
  • By directed line,do you mean about the vector that gets rotated during rotation? –  Jan 03 '18 at 14:16
  • that's what i mean –  Jan 03 '18 at 14:20
  • @GENESECT That is still baffling. If you mean the axis of the rotation, that line is always the same for every rotation: the normal to the plane... so what is the point of talking about it? If you are referring to a line in the plane... what line? All rays starting at the origin get rotated during a (nontrivial) rotation. – rschwieb Jan 03 '18 at 14:23
  • @rschwieb i don't get what u are saying.......can u tone that down for me and then explain my flaw.I am kinda new to this topic –  Jan 03 '18 at 14:29
  • @GENESECT We've already mentioned that the flaw is what you've said is vague and people can only guess what you mean. Now it's your turn to explain. Give an example of "two complex numbers and the line formed by them" and explain what "properties they share" and why you think they are "entirely different" – rschwieb Jan 03 '18 at 14:35
  • If you are ok, you can accept the answer and set as solved. Thanks! – user Jan 04 '18 at 19:59

2 Answers2

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The relation is that complex number can represent points in $\mathbb{R^2}$ since they obbey at the same algebraic rules valid for the vectors in $\mathbb{R^2}$.

Thus you are allowed to use complex numbers to solve geometric problems in $\mathbb{R^2}$ by algebraic manipulations.

user
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  • In addition, we may write $\mathbb{C}=\mathbb{R} \oplus \mathbb{R} i$. And for quaternion, $\mathbb{H}=\mathbb{C} \oplus \mathbb{C} j$ and can be used in relativity to represent four-vectors. – Ng Chung Tak Jan 03 '18 at 14:17
  • @NgChungTak Thanks, please add your comment about it as an answer, I'm not skilled about quaternions. – user Jan 03 '18 at 14:20
  • That'll be going too far. In short, the imaginary parts take care the vector in real space $\mathbb{R}^3$ while the real part take care the "time" as in relativity. The crucial things are $i^2=j^2=k^2=-1$, $i=jk$, $j=ki$, $k=ij$, $-i=kj$, $-j=ik$ and $-k=ji$ that take care the vector product. – Ng Chung Tak Jan 03 '18 at 14:30
  • can you teach me mathjax –  Jan 04 '18 at 20:05
  • it is simple once you learn the simbols to use, take a look here https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference – user Jan 04 '18 at 20:08
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You can think of a complex number $a+bi$ as a vector $$ (a,b) $$ in $\mathbb{R}^2$. They obeys all the familiar laws of vectors $$ \lambda(a,b)+\tau(c,d)=(\lambda a+\tau c,\lambda b+\tau d) $$

Note however that $\mathbb{C}$ has additional structure $\mathbb{R}^2$ does not. It has a funky form of multiplication of vectors, $$ (a,b)*(c,d)=(ac-bd,bc+ad) $$ which fits nicely with the observation that multiplication by $i$ is rotation by $90$ degrees, as representing $i$ in our notation by $(0,1)$ we have $$ (0,1)*(c,d)=(-d,c)=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\begin{bmatrix}c\\d\end{bmatrix} $$ the familiar matrix representing rotation by $90$ degrees.

operatorerror
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