1. 講者： 許永昌 老師 1

2. Contents Find the rotational axis of an UCM. Definition Levi-Civita Symbol: p153, Eq.2.133. Examples Summary 2

3. Find the rotational axis of an UCM (uniform circular motion) If a ball whose motion is an UCM is located at position r (related to the center of this circle) with velocity v at an instant. How to describe the direction of rotational axis? A: Usually we use the right hand rule to define the direction of rotational axis. However, it is not an algebraic definition. From the right hand rule, we get: r  v=  v  r= . r  r  3

4. Find the rotational axis of an UCM (uniform circular motion) For this purpose, we want this operation obeys If we want this cross product is a linear operator, we can require the cross product for 3D obeys We get 4

5. Cross Product |A  B|=ABsin . Because A  B=  B  A, we get A  B=A  (B  +B // ) =A  B  = ABsin  Ĉ. Levi-Civita Symbol:  ijk 5 A B  1  x 2  y 3  z 1  x 2  y 3  z

6. Usage Find a normal of a plane If I provide three points on a 3D plane. Find the rotational axis Get the area of the parallelogram Why? Get the volume How? Calculate the angular momentum L, torque , Lorentz force, … P22 6

7. The comparison of dot product and cross product Dot Product: A  BCross Product: A  B Symmetric: A  B=B  AAntisymmetric: A  B =  B  A ScalarVector A  B=AB // =A // B=ABcos .|A  B|=AB  =A  B=ABsin . AB=iAiBiAB=iAiBi http://en.wikipedia.org/wiki/Dot_prod uct http://en.wikipedia.org/wiki/Inner_pro duct_space http://en.wikipedia.org/wiki/Cross_pro duct http://en.wikipedia.org/wiki/Exterior_ product 7

8. Examples p24 ~ p25 The tangential velocity: The shortest distance between two lines Medians of a Triangle Meet in the center. Used to prove that c // m. 8 Why? Conditions? mm

9. Summary |A  B|=ABsin . Its magnitude is related to the concept of area and its direction is defined by the right hand rule. 9

10. Homework 1.3.4 1.3.7 1.3.11 10

11. Nouns 11