Ch1.4e (Ch1.5) Triple Scalar Product and Triple Vector Product

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Ch1.4e (Ch1.5) Triple Scalar Product and Triple Vector Product 講者：許永昌老師

Contents The angular momentum and the kinetic energy of a particle Triple Scalar Product A(BC) Triple Vector Product A(BC) Summary

L and K of a particle Assume a particle mass m whose angular velocity is w and its position is r at an instant. What is the angular momentum and the kinetic energy at this instant? L=rp=mr(wr)=mr2w –m(rw)r K= ½mv2=½m(wr)(wr) =½mr2w2-½m(wr)2 =½Izzw2. (if w//z)  Exercise 1.4.3 & 1.4.4. w r

Triple Scalar Product A(BC)= B(CA)= C(AB)= A(BC)= A(BC)= () volume of parallelepiped defined by A, B and C. Code: example_cross_product.m

Triple Vector Product (P28, P32e) No Commutative relation: A(BC) (AB)C. E.g. A(BC)=(AC)B-(AB)C. Proof: A(BC) should be perpendicular to A and BC, therefore, A(BC)=uB+vC and A(uB+vC)=0. Therefore, A(BC)=w( (AC)B-(AB)C ). w is independent of A (owing to A=SiAiûi), and is also independent of B and C.

Summary Because vectors are independent of the coordinates, a vector equation is independent of the particular coordinate system. The coordinate system only determines the components. Triple Scalar Product: V3V0 (scalar) A(BC)= B(CA)= C(AB) Triple Vector Product: V3V A(BC)=(AC)B-(AB)C

Homeworks 1.4.1e (1.5.2) 1.4.2e (1.5.3) 1.4.3e (1.5.5) 1.4.4e (1.5.6)