This is the second of three lectures on Hamilton's discovery of quaternions, and here we introduce rotations of three dimensional space and the natural problem of how to describe them effectively and compose them. We discuss the geometry of the sphere, take a detour to talk about composing planar rotations with different centers, talk about the connections between reflections and rotations, and introduce the basic algebraic framework with vectors, the dot product and the cross product. As in the first lecture, there is a lot of information here, so by all means take it slowly, and break it up by pausing and absorbing the ideas before going further.

Euler's theorem on the composition of rotations is an important ingredient. You will also learn that a curious addition of spherical vectors on the surface of a sphere provides an effective visual calculus for composing rotations.

This lecture prepares us for the next, where we introduce Hamilton's quaternions, which connect the dot product and cross product in a remarkable way, and yield probably the most effective current technique for managing rotations in graphics, video games and rocket science. So yes, this is really rocket science!

Video Contents:
00:00 Introduction to rotations and their composition-
03:25 Rotations of 3-Dimensional space ( geometrically )
09:38 Planar situation
14:25 Algebra of planar rotations
23:36 Rotation of 3D space as a product of 2 reflections
37:04 Algebra of 3D Rotations about 0
42:57 Euler theorem - The product of two rotations is a rotation
44:40 The analytic approach ( via Linear Algebra)
48:24 How to define 2 directions to be perpendicular; vectors
52:43 Cross product of 2 vectors

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Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.

My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/...

My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.

Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects!

If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at   / njwildberger   Your support would be much appreciated.

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Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained:    / playlist  
list=PL8403C2F0C89B1333
Year 9 Maths:    • Year9Maths  
Ancient Mathematics:    • Ancient Mathematics  
Wild West Banking:    • Wild West Banking  
Sociology and Pure Mathematics:    • Sociology and Pure Mathematics  
Old Babylonian Mathematics (with Daniel Mansfield):    / playlist  
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History:    • MathHistory: A course in the History ...  
Wild Trig: Intro to Rational Trigonometry:    • WildTrig: Intro to Rational Trigonometry  
MathFoundations:    • Math Foundations  
Wild Linear Algebra:    • Wild Linear Algebra  
Famous Math Problems:    • Famous Math Problems  
Probability and Statistics: An Introduction:    • Probability and Statistics: an introd...  
Boole's Logic and Circuit Analysis:    • Boole's Logic and Circuit Analysis  
Universal Hyperbolic Geometry:    • Universal Hyperbolic Geometry  
Differential Geometry:    • Differential Geometry  
Algebraic Topology:    • Algebraic Topology  
Math Seminars:    • MathSeminars  
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And here are the Wild Egg Maths Playlists:

Triangle Centres:    • ENCYCLOPEDIA OF TRIANGLE CENTERS  
Six: An elementary course in pure mathematics:    • Six: An elementary course in Pure Mat...  
Algebraic Calculus One:    • Algebraic Calculus One  
Algebraic Calculus Two:    • Algebraic Calculus Two  
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