chapter 3: vectors and coordinate systems€¦ · chapter 3: vectors and coordinate systems → the...

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3.1 Scalars and Vectors p. 72-73

3.2 Properties of Vectors p. 73-78

Chapter 3: Vectors and Coordinate Systems

3.3 Coordinate systems and vector components p. 78-82

3.4 Vector Algebra p. 82-85

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3.1 Scalars and Vectors p. 72-73


3.2 Properties of Vectors p. 73-78

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In Phys 221 we will encounter two fundamentalmathematical quantities to describe nature: scalars and vectors

Formally this classification refers to how the quan-tities change when you change your coordinatesystem, but this is not important for Phys 221.More such quantities do exist (tensors, spinors,pseudo-scalars ...)

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Phys 221 definition:

A scalar is a quantity which is represented by asingle number

A vector v has a magnitude (which is a scalar, i.e.,a number) and a direction

Often one denotes themagnitude of the vectorwith the same symbol, butwithout an arrow:v = magnitude of v

→

→

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Two vectors are the same when they have the (I) same magnitude and the (ii) same direction. Itdoesn't matter at which position,in your coordinate system, theirtails are drawn.

x

y

equal vectors

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An important operation: multiplication of a vectorwith a POSITIVE scalar (greater than 0)

This changes the length, but not the direction of avector:

v = ( v, direction) ⇒ 5 v = (5 v, direction)

The direction is often indicated by an angle θ

⇒

→

→

What happens if c=1?No change in magnitudeNo change in direction

The vector remains the same!

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Another important operation: multiplication of avector with a NEGATIVE scalar (smaller than 0)

This changes the length, but also the direction of avector:

v = ( v, direction) ⇒ 5 v = (-3 v, direction)

The direction is often indicated by an angle θ

⇒

→

→

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Multiplying with -1.

⇒

What happens if c=-1?No change in magnitudeBUT change in direction

The vector is not the same!

multiplying with zero (or c=0):

0.v = 0 with thezero vector 0

The zero vector doesn'thave a certain direction.Intuitively because “0”means that we are notpointing in anywhere.

TWO SPECIAL CASES Multiplying with 0.

→ →→

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Remember that theNegative vector isused to subtract 2vectors:

P – Q = P + (-Q)

⇒

→ → → →

Which figure shows

Cliction 3.1

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Recall vector addition:

⇒

Which figure shows

Cliction 3.2

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3.3 Coordinate systems and vector components p. 78-82

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The coordinate description of vectors makesvector calculations much easierBasic idea: dealing with orthogonal vectors is sim-ple, dealing with non-orthogonal vectors is NOT!!

Once we have choosen ourcoordinate axes we can ex-press every vector A as asum of two vectors Ax andAy that a parallel to theseaxes: A = Ax + Ay

⇒

→→→

→ → →

NOTEA and A without the arrows are simply

called COMPONENTS.x y

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A conventional Cartesian coordinate system

⇒

The POSITIVE y-axis is located 90 counterclockwise (ccw) from the

POSITIVE x-axis.

Quadrants I through IV counterclockwise (ccw)

o

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The magnitude of Ax and Ay determines (up to thesign) the components of A along the x and ydirection

⇒

→ →

→

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→ ⇒

What are the x- and y-components Cx and Cy of vector

1) Cx= –3 cm, Cy = 1 cm2) Cx= –4 cm, Cy = 2 cm3) Cx= –2 cm, Cy = 1 cm4) Cx= –3 cm, Cy = –1 cm5) Cx= 1 cm, Cy = –1 cm

Cliction 3.3

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The components can be used to determinemagnitude and direction (angle) of a vector

Pythagoras ⇒

Trigonometry ⇒

⇒

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Using the IV quadrant:

Trigonometry ⇒

Pythagoras ⇒

⇒

C = C.sin( )

C = - C.cos( )

x

y O

O

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→ ⇒

Angle that specifies the direction ofis given by

1) tan–1(Cx/Cy)2) tan–1(Cx/|Cy|)3) tan–1(|Cx|/|Cy|)4) tan–1(Cy/Cx)5) tan–1(Cy/|Cx|)

Cliction 3.4

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End of Week 3


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3.4 Vector Algebra p. 82-85

IMPORTANT

Our choice of a coordinate system may be arbitrary, but once we decideto place a coordinate system on a problem we NEED something to tell

us ``THAT direction is the positive x-direction”.This is what the unit vectors do which are CRUCIAL in vector algebra.

WITHOUT UNIT VECTORS, VECTOR ALGEBRA IS MEANIGLESS!!!

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The sign of the components of A becomes mostobvious if we introduce unit vectors i and j toindicate the positive direction of the coordinateaxes:

We then can write anyvector A as

The components Ax and Aycan be positive, negative,or zero, and carry the unit (e.g., m/s) of the vector

⇒

ˆ ˆ

→

→

Read “i hat” and “j hat”

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→ ⇒

The FULL DECOMPOSITION of a vector is

Recall the rule for adding vectors:

A→

xA→

y

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Thus, every vector A is uniquely determined by thepair of numbers (Ax , Ay). Adding two vectors now becomes very easy.

Say we start with A = (A , A ) and B = (B , B ) andadd them …

⇒

→

… is equivalent to …

… which implies…

→ →yx xy

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This means when we add vectors we simply addtheir x-components and their y-components. Thesecomponents are added like ordinary numbers andcan be positive or negative.

Subtraction works likewise:

⇒

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As a practise now add three vectors:

⇒

D = A + B + C→ → → →

… and show that the components of D are→

D = A + B + CD = A + B + C

XXXX

Y Y YY

Now try to guess the components of:E = A + B - C

F = D - E - 2.C

G = 10.F + 2.D - E - 3.C

→ → → →

→ → → →

→ → → → →

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Multiplication with a scalar c also simply amountsto multiplying the components with c:

In summary, vector algebra is just like normalalgebra done with each vector component.Trigonometry helps to find magnitude and direction

⇒

Note: this means scalar “c”multiplied by the vector S→

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The direction of a vector is a definite, well definedquantity. However if we change the coordinatesystem, the direction relative to the coordinateaxes will change; this is the formal definingproperty of a vector

⇒

x

y

x'

y'

Same red vectors, but different orientation relative to the two differentpairs of coordinate axes.

TILTED AXES

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Note that the unitsvectors are also tilted.

⇒

TILTED AXES

A = A = A.cos( )

A = A = A.sin( )

x

y

A = A + A→ → →

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→ ⇒

TILTED AXES: example

Finding the component of a force F perpendicular to a surface.→

F = F.sin(20) = 10N.sin(20)=10N.(0.342)=3.42 No o

F→

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Chapter 3Reading Quiz

Chapter 3: Vectors and Coordinate SystemsPh

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What is a vector?

1) A quantity having both size and direction2) The rate of change of velocity3) A number defined by an angle and a magnitude4) The difference between initial and final displacement5) None of the above


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What is the name of the quantity represented as

1) Eye-hat2) Invariant magnitude3) Integral of motion4) Unit vector in x-direction5) Length of the horizontal axis


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This chapter shows how vectors can be added using

1) graphical addition.2) algebraic addition.3) numerical addition.4) both 1 and 2.5) both 1 and 3.


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To decompose a vector means

1) To break it into several smaller vectors.2) To break it apart into scalars.3) To break it into pieces parallel to the axes.4) To place it at the origin.5) This topic was not discussed in Chapter 3.


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Selected Problems

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End of Chapter 3

IMPORTANT:

Print a copy of the SUMMARY page (p. 86)and add it here to your lecture notes.

It will save you crucial time when trying to recall:Concepts, Symbols, and Strategies


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chapter 3: vectors and coordinate systems€¦ · chapter 3: vectors and coordinate systems → the...

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