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Ch1.3 Vector or Cross Product

講者： 許永昌 老師

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ContentsFind the rotational axis of an UCM.DefinitionLevi-Civita Symbol: p153, Eq.2.133.ExamplesSummary

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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:rv=-vr=w.

rv

w

r

v

w

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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

ˆ ˆ ˆˆ ˆ ,

ˆ 0,

ˆ 0.

A B B A C

C A

C B

& 0,

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆRight hand rule, i.e. , and .

A B B A C C A C B

A B D A B A D

x y z y z x z x y

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ.

x y z x y z

x x y y z z

x y y x y z z y z x x z

y z z y z x x z x y y x

A A A B B B

A B A B A B

A B A B A B A B A B A B

A B A B A B A B A B A B

A B x y z x y z

x x y y z z

x y y z z x

x y z

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Cross Product|AB|=ABsinq.

Because AB=-BA, we get AB=A(B+B//) =AB= ABsin q Ĉ.

Levi-Civita Symbol: eijk

AB

B

B//

q

, ,

ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ

y z z y z x x z x y y x x y z

x y z

ijk j ki j k

A B A B A B A B A B A B A A A

B B B

A B

x y z

A B x y z

i1 x2 y3 z

123 231 312

132 213 321

112

1,

1,

otherwize 0 (E.g. =0).ijk

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UsageFind a normal of a plane

If I provide three points on a 3D plane.Find the rotational axisGet the area of the parallelogram

Why?Get the volume

How?Calculate the angular momentum L, torque t,

Lorentz force, …P22

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The comparison of dot product and cross productDot Product: AB Cross Product: AB

Symmetric: AB=BA Antisymmetric: AB = -BA

Scalar Vector

AB=AB//=A//B=ABcosq. |AB|=AB=AB=ABsinq.

AB=SiAiBi

http://en.wikipedia.org/wiki/Dot_producthttp://en.wikipedia.org/wiki/Inner_product_space

http://en.wikipedia.org/wiki/Cross_producthttp://en.wikipedia.org/wiki/Exterior_product

ˆ ˆ ˆ

for 3D.x y z

x y z

A A A

B B B

x y z

A B

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Examples p24 ~ p25The tangential velocity:

The shortest distance between two lines

Medians of a Triangle Meet in the center.Used to prove that c // m.

ˆ ˆˆ ˆ ,

ˆ .

r t

t

d drr v v

dt dt

r

rv r r

v r Why? Conditions?

m

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Summary

|AB|=ABsinq.Its magnitude is related to the concept of

area and its direction is defined by the right hand rule.

ˆ ˆ ˆ

for 3D.x y z

x y z

A A A

B B B

x y z

A B

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Homework1.3.41.3.71.3.11

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NounsVector or Cross ProductLevi-Civita Symbol : p153, Eq.2.133.

http://en.wikipedia.org/wiki/Levi-Civita_symbol

Right-hand rule: P63