Tagged Questions
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1answer
51 views
Two quick eigenvalues & complex numbers questions
A) For a vector $v\in\mathbb{C^n}$, is $Im(-v)=Im(\overline{v})$ ?
($Im(v)$denoting the imaginary part of the vector $v$)
My understanding: since every row of the vector is a complex number (say ...
3
votes
1answer
64 views
Does anyone know any resources for Quaternions for truly understanding them?
I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
0
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1answer
29 views
Need help interpreting an equation from an article (related to quaternions).
At this link, about half way down the page, there is an equation I don't understand
http://physicsforgames.blogspot.com/2010/02/quaternions-why.html
This is the equation.
$$VV† = -x^2I^2 - y^2J^2 - ...
4
votes
3answers
89 views
How do you construct the quaternion and the multiplication rules, like Hamilton did?
So, I understand complex number multiplication, and how it represents $2D$ rotations.
What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
3
votes
1answer
77 views
The multiplication of 2D vectors produces what?
I am trying to learn about rotation quaternions, and in the process I am currently looking at 2D vector multiplication.
To avoid confusion with other types of multiplication, this is the basic form I ...
9
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3answers
212 views
square root of $1/2 + \sqrt3/2?$
Playing with Maple, I noticed that it gives the square root of $c = 1+\frac{\sqrt3}{2}$ as equal to $a = \frac{1}{2}+\frac{\sqrt3}{2}$.
Indeed it checks out. But I got curious: how can I find that ...
2
votes
1answer
37 views
Bound on unit vectors
could someone help me with this simple problem. As always with homework, hints are specially welcome.
Let $v=(v_1,v_2)$ be a two-dimensional unit vector with complex coefficients. If $|v_1|<a$ and ...
2
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2answers
93 views
Multiplying Complex Numbers by i
But I am wondering why isit $PQ \perp QR$ and not $QP \perp QR$ as shown below?
UPDATE
How do I get the equation: $(i-1)b=ic-a=i(1-2i)-(-1+4i)=3-3i$?
Where does $(i-1)$ come from? I dont ...
5
votes
1answer
204 views
Is there a “good” way to visualize complex vectors?
We often represent complex numbers as vectors in $\mathbb{R}^2$ with $x$ being the real axis and $y$ being the imaginary axis. We often represent 2-dimensional vectors over $\mathbb{R}$ in a similar ...
3
votes
1answer
186 views
The real part treated like an angle in complex vector spaces
In my current lecture I regularly encounter usage of the real part of, say, a scalar product of two vectors similar to angles in classical geometry. For example in Hilbert space theory:
Let $H$ be a ...
4
votes
3answers
422 views
What's the Difference Between a Vector and an Hypercomplex Number?
What's the difference between a vector and an hypercomplex number? For instance a 4-vector and a quaternion. They seem to share many properties.
Perhaps this question could be put more generally as: ...
2
votes
2answers
745 views
What are the rules for complex-component vectors and why?
I want to take the inverse of a dot product, where both vectors have complex components. In other words, if $\textbf{A} \cdot \textbf{B} = d$, and I know $\textbf{A}$ and $d$, I want to find a ...
1
vote
3answers
514 views
Geometric interpretation of the multiplication of complex numbers?
I've always been taught that one way to look at complex numbers is as a cartesian space, where the "real" part is the x component and the "imaginary" part is the y component.
In this sense, these ...
