Example of Quaternions

Matrix Theory: For the quaternion alpha = 1 - i + j - k, find the norm N(alpha) and alpha^{-1}. Then write alpha as a product of a length and a direction.

[MathDoctorBob]

You're welcome, and thanks! As noted below, a small playlist on quaternions is on the wishlist for 2013.

[peterhunt]

Like when he is carrying that stick you really don’t feel like being cheeky in his class.

[rtuure]

This is very interesting stuff, Quaternion algebra was introduced by Irish mathematician Sir William Rowan Hamilton, at his time it was a revolution, excellent video

[MathDoctorBob]

Yes, and it does. You can relate products of quaternions to the dot and cross products on R^3. This is formalized nicely in the language of Clifford algebras.

Even better, unit quaternions can be used to represent rotations in R^3. For a rotation, we need an axis of rotation and angle of rotation in the plane. The imaginary part sets the axis of rotation, and, with that, the real term sets the angle.

Quaternions are #3 for requests. One day I'll get more videos up on them.

[MathDoctorBob]

Yes, it's a bit abstract, but if you let the unit quaternions act on the imaginary quaternions (three-space) by conjugation, each rotation matrix can be represented by two unit quaternions.

Here's the hierarchy: real to complexes (lose total ordering), complex to quats (lose commutativity), quats to octonions (lose associativity), and possibly octs to sedonions or exceptional Lie algebras. - Bob

[MathDoctorBob]

I've been tenure track twice (long story), went corporate as an actuary, and now teaching part-time at CUNY (NYC College of Tech). I was going to wait to search for full-time next year, but there's a chance something could happen this cycle.

[Titurel]

Thanks Dr Bob! I was reading Penrose's Road to Reality trying to understand this and was confused. You made everthing perfectly clear!

[omgitzsteg]

Thanks for your explanation on quaternion math, very clear, concise and completely makes sense of a complicated, if not antiquated subject. Please consider also doing a video on the "point" of quaternions to begin with. That would be awesome!

[MathDoctorBob]

@odinheim The last board is quaternion multiplication - the whole i.j = k business. I check that alpha*alpha^{-1} = 1 because I need to give an example of multiplication with linear combinations.

At some point, I'll do more with H and make the connection to the dot and cross products on R^3 explicit.

[omgitzsteg]

Good luck. Your presentation is sharp and somehow manages to make even this topic interesting. A rare talent in your field!

I'll keep checking out your other videos, as well.

[MathDoctorBob]

Antiquated, but still vital. As noted below, a quaternion mini-series is on my to-do list. The plan is to finish Galois Theory and start a Real Analysis series. Quaternions will make for a good break from analysis.

[Sam]

concise and on point. much appreciated Sir.

[MathDoctorBob]

You'd be surprised. If you work with rotations of three-space (think video games/CGI), quaternions are often easier to work with than matrices. In fact, much physics was developed using quaternions before the vector concept was formalized. - Bob

[MathDoctorBob]

Yes. If you follow the link to my site, Problem Set 9 in Group Theory has a section on quaternions with solutions. - Bob

[alexeykletnoy]

Thanks Bob, I was trying to find an easy way to explain this to my friend. Good vid, thumbs up.

[SpazzyMcGee]

I'm a physicist trying to understand quaternions. Are the imaginary numbers i, j, and k the same thing as the unit vectors i, j, and k I used for three dimensional space? If so, then what is the physical significance of the real component, or does it not have any physical significance?

[MathDoctorBob]

@Trotskisty Old sparring stick. I've upgraded to a regular pointer. - Bob

[MathDoctorBob]

@Trotskisty Old sparring stick, but I use something more pointer-like these days. - Bob

[alfonshomac]

is it possible to see a derivation or construction of all these artifacts that are flying at my head right now? I don't like memorizing as much as understanding.

[rtuure]

sorry were you doing the dot product at 4:50 on the video ? alpha*alpha^-1 ?