Gaddamit Euler, I just discovered this this morning, I was actually writing this down as a way to address some famous Euclid problem on numberphile. So basically we are creating dimensions on paper, awesome lol.
@24:40 - "In general" formula - are b,c and d measured in radians? Somehow I don't think they can be. If a=0 and |v|=1 it is a pure quaternion. Then cos(|v|)=0 and sin(|v|) should be 1. That occurs at pi/2. so |v| = pi/2. But false because |v| = 1. Maybe it should be written cos(pi/2*|v|) and same for sine? Or maybe sin() and cos() are operating on right angle measures. Which would be so wonderful, since given angle x, powers of i^x breaks into sin(x) and cos(x) angles directly and without pi. I think I am wrong, where am I wrong?
Your videos on Quaternions are exceptional. Thank you. Question: You clearly relate vector part "bi + cj + dk" to 3D space. What geometrical interpretation does the scalar component have?
If you want to think of a quaternion in a geometrical sense, you can think of them as 4 dimensional numbers. I highly recommend you watch 3blue1brown's videos on quaternions for this intuition if you are interested in their spatial interpretation.
i suggest you start with q=a+b1*i+b2*j+b3*k and b=sqrt(b1^2+b2^2+b3^2) then q=a+b*Î with î the specific normalized quarternion here with |î|=1 and î=b1/b*i+b2/b*j+b3/b*k, then e^q becomes very simular to the complex case: q=a+b*î and e^q=e^a*(cos(b)+sin(b)*î) which notation seems much more alike that in the complex situation. Right? look at Michael Penn's way of doing this, i suggested there the same way of writing it down.
So it appears that one can manipulate numbers and define their operations arbitrarily to satisfy the goals one seeks. I am obviously not a mathematician, but it would appear one can define number operations depending in which dimension they operate. Are their corresponding rules for each dimension? Or was this just a "fix" to facilitate a mathematical way of describing rotations in three dimensions? I would be grateful of any insight and clarification. Thanks.
+Barry Hughes To be able to do 3D rotations and generalize the complex numbers was certainly the goal behind quaternions. The tough part was discovering the right definition of the way the imaginary units i,j,k should be multiplied to do this correctly. Of course, you could choose different definitions and different properties would result, just not the one you would be interested in for 3D rotations. I disagree that this is a fix/kludge for doing 3D rotations, in that it is more natural than other ways of doing rotations like with cross products. The cross product, I think, is a kludge and is unnatural. For rotations, an even more natural system is provided by geometric algebra (GA), where the quaternions pop up for free in 3D. Furthermore, the GA approach easily deals with higher dimensional rotations without special tricks for each particular dimension.
Does e^(ai+bj+ck) = e^(ck+bi+ai) ? I think so, but in another way I don't since the order of rotations is important. Like how could e^ai × e^bj × e^ck == e^ck × e^bi × e^ai when the order of rotations are different. It must be different. But then I know the ai+bj+ck == ck+bj+ai. So how could e to the power of the same thing be different?
In your example of plugging 'ix' into the exponent function: exp(ix), I notice that you do not take the length of 'ix' to be |x|, as it technically is. It seems like your derivation should be exp(ix) = cos|x| + i*sin|x|*(x/|x|), which would then require showing that the imaginary part is indeed equal to i*sin(x). It is equal, but this doesn't seem like a step you can skip without losing rigor. (cos|x| = cos(x) by definition that cosine is an even function: cos(-x) = cos(x), so the Real part of the result is more obvious.)
Err, I should point out the youtube strike thru thing is also causing a couple signs to go missing (first sqrt(-7), 33/37 and there is suppose to be a space in there (between 37 and (43/259)). The distance of that vector should be 1. Anyway, do you understand my confusion? Maybe I should have been more specific. The sqrt(neg) puts an i out in front, i*sqrt(7). I suppose you would put an i in front of that and expand i*(33/37+neg(51/259)*sqrt(neg7)) = (51/259)*sqrt(7)+(34/37)*i. And just change up the i with the j and k on the other two. I guess I asked a question without putting much thought into it.
Thomas Olson you cant write it as a vector like this. you have a vector that is purely Real (thus 3dimensional) which can be expressed as a Quaternion, with the three components representing the i,j,k part. But in your case you have a vector with complex components, making it at least 6 dimensional, which can not be expressed as a Quaternion
Superb! All your videos about quaternions and their manipulation are the best ones I've seen so far on TH-cam! Amazing work!
This was very helpful for understanding how Euler's Formula applies to quaternions. Thank you very much for making this!!
Gaddamit Euler, I just discovered this this morning, I was actually writing this down as a way to address some famous Euclid problem on numberphile. So basically we are creating dimensions on paper, awesome lol.
Perhaps Euler's equation becomes slightly more nice when we set v=vbarvhat (see the video at 17:44)
The first two exercises really blew my mind!
The key is that "a" commutes with "bi + cj + dk", but "i" doesn't commute with "j".
@24:40 - "In general" formula - are b,c and d measured in radians? Somehow I don't think they can be.
If a=0 and |v|=1 it is a pure quaternion. Then cos(|v|)=0 and sin(|v|) should be 1. That occurs at pi/2. so |v| = pi/2. But false because |v| = 1.
Maybe it should be written cos(pi/2*|v|) and same for sine?
Or maybe sin() and cos() are operating on right angle measures. Which would be so wonderful, since given angle x, powers of i^x breaks into sin(x) and cos(x) angles directly and without pi.
I think I am wrong, where am I wrong?
Best explanation so far
Your videos on Quaternions are exceptional. Thank you.
Question: You clearly relate vector part "bi + cj + dk" to 3D space. What geometrical interpretation does the scalar component have?
The vectorial part of a Quaternion is no more that rotational planes from Clifford Algebra. They are no 3D Space.
If you want to think of a quaternion in a geometrical sense, you can think of them as 4 dimensional numbers. I highly recommend you watch 3blue1brown's videos on quaternions for this intuition if you are interested in their spatial interpretation.
i suggest you start with q=a+b1*i+b2*j+b3*k and b=sqrt(b1^2+b2^2+b3^2) then q=a+b*Î with î the specific normalized quarternion here with |î|=1 and î=b1/b*i+b2/b*j+b3/b*k, then e^q becomes very simular to the complex case: q=a+b*î and e^q=e^a*(cos(b)+sin(b)*î) which notation seems much more alike that in the complex situation. Right? look at Michael Penn's way of doing this, i suggested there the same way of writing it down.
Incase i have q=(5,1,1,1) then what is q^2 ?
Brilliant you do a fantastic job explaining the math.. I am learning more than a college class on this subject
Are Pauli matrices, i,j,k or bivectors
You helped us out a lot, nice job :)
You are welcome. I hope you found it useful.
UKelectro m
So it appears that one can manipulate numbers and define their operations arbitrarily to satisfy the goals one seeks. I am obviously not a mathematician, but it would appear one can define number operations depending in which dimension they operate. Are their corresponding rules for each dimension? Or was this just a "fix" to facilitate a mathematical way of describing rotations in three dimensions? I would be grateful of any insight and clarification. Thanks.
+Barry Hughes
To be able to do 3D rotations and generalize the complex numbers was certainly the goal behind quaternions. The tough part was discovering the right definition of the way the imaginary units i,j,k should be multiplied to do this correctly. Of course, you could choose different definitions and different properties would result, just not the one you would be interested in for 3D rotations.
I disagree that this is a fix/kludge for doing 3D rotations, in that it is more natural than other ways of doing rotations like with cross products. The cross product, I think, is a kludge and is unnatural. For rotations, an even more natural system is provided by geometric algebra (GA), where the quaternions pop up for free in 3D. Furthermore, the GA approach easily deals with higher dimensional rotations without special tricks for each particular dimension.
Most grateful for your reply, and thanks.
It's all just Geometric Algebra
I’m interested in the image that appears in the main listing of this video and its explanation. Do you have a video for this?
Does this mean that the exponential of the zero quaternion is undefined? Or do we have to take the limit approaching zero to get 1?
Does e^(ai+bj+ck) = e^(ck+bi+ai) ? I think so, but in another way I don't since the order of rotations is important.
Like how could e^ai × e^bj × e^ck == e^ck × e^bi × e^ai when the order of rotations are different. It must be different.
But then I know the ai+bj+ck == ck+bj+ai. So how could e to the power of the same thing be different?
Self answer: The first line is true. They are equal. Saying they are equal to something like e^ai × e^bj × e^ck isn't true.
In your example of plugging 'ix' into the exponent function: exp(ix), I notice that you do not take the length of 'ix' to be |x|, as it technically is. It seems like your derivation should be exp(ix) = cos|x| + i*sin|x|*(x/|x|), which would then require showing that the imaginary part is indeed equal to i*sin(x). It is equal, but this doesn't seem like a step you can skip without losing rigor. (cos|x| = cos(x) by definition that cosine is an even function: cos(-x) = cos(x), so the Real part of the result is more obvious.)
+HeavyMetalMouseYeah I skipped that step.
tome su like buen hombre!
+Jonathan Salazar¡Gracias!
How would you go about playing with this vector?
< 34/37+neg(51/259)*sqrt(neg 7) | neg33/37+neg(43/259)*sqrt(neg7) | 7/37+(45/259)*sqrt(neg7) >
+Thomas Olson
I have no idea.
Err, I should point out the youtube strike thru thing is also causing a couple signs to go missing (first sqrt(-7), 33/37 and there is suppose to be a space in there (between 37 and (43/259)). The distance of that vector should be 1.
Anyway, do you understand my confusion? Maybe I should have been more specific. The sqrt(neg) puts an i out in front, i*sqrt(7). I suppose you would put an i in front of that and expand i*(33/37+neg(51/259)*sqrt(neg7)) = (51/259)*sqrt(7)+(34/37)*i. And just change up the i with the j and k on the other two.
I guess I asked a question without putting much thought into it.
Thomas Olson you cant write it as a vector like this. you have a vector that is purely Real (thus 3dimensional) which can be expressed as a Quaternion, with the three components representing the i,j,k part. But in your case you have a vector with complex components, making it at least 6 dimensional, which can not be expressed as a Quaternion
Now please do for 2^(n-1) dimensional Cayley-Dickinson Construction algebras.
Excellent
Thank you
i was just searching for this to use it in my statics course hehehehehe
ok now im mind blown. Where have I always been
Ian 216 up I the air here.
Sorry. Thankyou for your attention to others.
Thanks.
+Márcio Laubstein You are quite welcome.
Handy.