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Ch 3 Vectors and Coordinate Systems

講者： 許永昌 老師

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ContentsGeometric representation

Vector additionMultiplication by a scalar

Coordinate representationDecomposition and components

Vector AlgebraUnit vector

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Geometric representation ( 請預讀P72~P77)

1. Vector: A quantity has a magnitude and a direction.

E.g.

2. Vector addition: tip-to-tail method.

Multiplication by a scalar: Zero vector: Negative vector:

200 ft, northeastS

A

B

A B

0 0, direction undefined

,direction opposite A A A

,direction of A A A

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QuizDraw the vectors:

A+BA-BA+2B

Properties:A+B=B+A(A+B)+C=A+(B+C)

A

B

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HomeworkStudent Workbook: P3-1~P3-2

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Coordinate representation ( 請預讀P78~P81)

Coordinate system:Origin.Orientation of axes.Cartesian coordinate system( 直角坐標系統 ):

Originx axis

y axis

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Decomposition and ComponentsDecomposition:

Components:Ax=Acosq. Ay=Asinq.

NOTE:當用到角度時，請務必畫出圖

來表示，不標的話，考試時會被扣分。理由：比方 pendulum 的角度，

就不同於這裡的。

Origin

x axis

y axis

x yA A A

A

xA

yA

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ExercisesDraw the position vector of point A. What are

the magnitude and direction of this vector?

Finding the components of B and C (unit: m).

x [m]

y [m]

x [m]

y [m]

-2-4

B C

A

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HomeworkStudent Workbook: P3-2~P3-3

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Vector Algebra( 請預讀 P82~P86)

Definition of unit vector: A vector whose magnitude is 1.î ( 1, positive x-direction )ĵ ( 1, positive y-direction )

Decomposition of a vector:

Therefore, D=A+B+2C is equal to

component vector component*unit vector

ˆ ˆ , .x y x y x yA A A A i A j A A

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2x x x x

y y y y

D A B C

D A B C

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HomeworkStudent Workbook: P3-3~P3-4.

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SummaryGeometric representation:

(magnitude, direction): cylindrical coordinate.

Coordinate representation:A=Axî + Ayĵ.A=(Ax , Ay).

The relations:

A

xA

yA

2 2

1

,

tan In the I & IV quadrants

x y

y

x

A A A

A

A

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ExercisesFind the components of B whose magnitude

is 2 m/s.

If the velocity of a particle located at r=î+ĵ m is v=3î-4ĵ m/s, please represent this situation geometrically.注意，單位不同。

x [m]

y [m]

-2-4

B

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ExerciseFind the components of this vector parallel

and perpendicular to this tilted line.

o30

3 units

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HomeworkTextbook: 3.18, 3.25, 3.28, 3.43一樣，做 terms and notation 的紙卡，自行練習。