vectors (physics)

8/7/2019 Vectors (Physics)

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VectorsVectorsVectorsVectors


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Sign Conventions

Its importantto haveaconsistentsetofalgebraicsignsto indicate direction forvectors

Why do we need them?A (+) ora(-) willtell usthe directionthe vector is going.

Sign conventions:(+): vector pointsright orup.

(-): vector pointsleft ordown.


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Sign Conventions

Sign conventions:(+): vector pointsright orup.

(-): vector pointsleft ordown.


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Topic:Vector Addition

In red isa displacement

vector.

Displacement distance& direction from theorigin/reference.

Vectorsare usually represented by alinewith an arrowhead atoneend whichindicatesthe direction.

The displacementabove is 8 blocks

North (magnitudeand direction)


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Ifthe person now walks 4 blocks South,what isthe new displacement?


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Tofigurethisout, wemust use vector

addition:+ = ?


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addition:+ = ?


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addition:+ =

This method iscalled the tip totailmethod. The new displacement is4 blocksNorth.


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Note how the displacement vector isdrawn:Itstartsatthetailofthefirst vector and endsatthetip ofthelast

vector.


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Topic:Vector AdditionLets gobacktoouroriginalexample: personstanding 8 blocks North. What ifshe walks 5blocks East. What isthe new displacement?What istheresultant?


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The direction is indicatedby the purple vector.

How aboutthe

magnitude?


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A2 + B2 = C2


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Topic:Vector AdditionTo summarize:

1. If we haveparallel vectors, then wecan addthem algebraically, using thetip totail method.

2. a. If we havevectors thatareatrightangles,

we draw the vectorstip totail.b. We draw theresultant to getthe direction.

c. We usethePythagorean theorem to getthe

magnitudeoftheresultant.These methods workbecause wecan moveavectorsaround aslong asthelength and directionstay thesame!


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Subtracting VectorsSubtracting VectorsSubtracting VectorsSubtracting Vectors

A B = ?


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Example Problem

What is 30 m/s north minus 15 m/seast?


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Example Problem


Question:

How many vectorsarethere in this problem?


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Example Problem


Question:

How many vectorsarethere in this problem?


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Words Math

What is30 m/s north minus15 m/seast?

north30As

m!X east15Bs

m!X


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north30As

m!X

Mathematically

east15Bs

m!X

A B = ?

Words Math


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north30As

m!X

Mathematically

east15Bs

m!X

A B = ?

Words Math

The Problem: Weonly know how toadd vectors.


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north30 sm

!

Xeast15B s

m

!

X

TheTrick:

Make it AdditionA B = A +( B)


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north30 sm

!

Xeast15B s

m

!

X

TheTrick:


Guess: what is B?


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Time Out!

C

This is C


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Time Out!

C

Point in the direction of C!


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Time Out!

C



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Time Out!

C


- C


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Continue


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north30 sm

!

X

east15B sm

!

X

TheTrick:


- B isB in theopposite direction


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north30 sm

!

X

east15B sm

!

X

TheTrick:


A B = A +( B)

- B isB in theopposite direction


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Graphical Addition of

Vectors

1. Makeascale drawing.2. Movethe vectorstip-

to-tail.3. Draw theresultant

vector (freetailtofreetip).

4. Measurethelength

oftheresultant witharuler.5. Measurethedirection oftheresultant with a

protractor.

A +( B)

1)

Which aretip-to-tail?

3)

2)

4)


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Vectors




4. Measurethelength


protractor.

A +( B)

1)

Which aretip-to-tail?

3)

2)

4)


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Vectors




4. Measurethelength


protractor.

A +( B)

1)

Draw theresultantfor 1):

from point ___to point ___?

2)

c

ba


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Vectors




4. Measurethelength


protractor.

A +( B)

1)



2)

c

ba

ac


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Vectors




4. Measurethelength


protractor.

A +( B)

1)



2)

f

d

e


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Vectors




4. Measurethelength


protractor.

A +( B)

1)



2)

f

d

e

f d


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Vectors




4. Measurethelength


protractor.

A +( B)

63


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Vectors




4. Measurethelength


protractor.

A +( B)

33.5 m/s @ 63 north ofwest

1 cm = 5 m/s

63

Yourfinalanswer!


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Motion Notes

(Continued):10.Subtracting Vector a.The negative sign makes one vectorpoint in

the opposite direction.

i.Ex: BA XX X has the same magnitudeasX

and the opposite direction

b.Deal with the directions, then add the

vectors.i. )B(ABA

XXXX!


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Topic:Vector Addition IIAnother way of determining theresultof

adding 2 vectors (i.e. resultant), istheparallelogram method.

This method will workeven ifthe vectors

are notatrightanglesRightangle Otherangle


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Topic:Vector Addition IIAnother way of determining theresultof

adding 2 vectors (i.e. resultant), istheparallelogram method.

This method will workeven ifthe vectors

are notatrightanglesRightangle Otherangle


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Topic:Vector Addition IINotice how I have drawn these vectorsto do

theaddition, tailtotail Toadd the vectors, draw a parallelogram (ashape in which oppositesidesare parallel) by

drawing parallel vectorsshown as dashed linesbelowRightangle Otherangle

Thetailsare drawntogether



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Topic:Vector Addition II Theresultant isthe diagonalofthe

parallelogram, shown in purplebelowForthedirection oftheresultant, measurewith a protractor. Forthemagnitude ofthe

resultant, measure with aruler.Rightangle Otherangle




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Topic: ComponentsofVectors Sometimes we need tochangeasingle

vector intotwoequivalent vectorsatrightanglestoeach other. Theequivalent vectorsarecalled components.

Any vector

ca

nbe

resol

ved

in

tocomponents. This process iscalled vector

resolution. Every vector will haveacomponentalong the

x-axisand the y-axis.


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Topic: ComponentsofVectors Suppose we havea vectorV in acertain

directionDraw horizontalandverticallinesfrom theta

ilof

th

ev

ectorto

makean xand y axis.

V

y

x


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Topic: ComponentsofVectors Imagine you areshining aflashlight downward

on the vectorfrom atop ofthe y-axisYou can imaginethatthiscreatesashadow on thex-ax

is

Thelength oftheshadow onthex-axis isthex

component (horizontalcomponent) ofthe vector.

V

Vx


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Topic: ComponentsofVectors Now shineaflashlighttowardsthe vectorfrom thex-axis

Similarly you can imaginethatthiscreatesashadow on the y-axis

V

Thelength oftheshadow on they-axis isthey-component

(verticalcomponent)ofthe vector

Vy


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Topic: ComponentsofVectorsBelow arethex & y componentsofthe

vectorV,

Vx = horizontalcomponentVy = verticalcomponent

V

Vy

Vx


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Topic: ComponentsofVectorsTry these:

A

B


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Topic: ComponentsofVectorsTry these:

A

Ax

Ay

B

Bx

By


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Polar Coordinate System

(Direction and Magnitude)


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Polar Math

(Cartesian)2 2

1

os

sin

tan

r x y

x r

y r

y

x

UU

U

!

!!

! x

yr


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Vectoraddition TwoWays:

1. Graphically: Draw vectorsto

scale, Tip toTail, and theresultant isthestraightlinefromstarttofinish

2. Mathematically: Employ vectormath analysistosolvefortheresultant vector


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Graphically 2-D Right A = 5.0 m @ 0 B = 5.0 m @ 90 SolveA+B

R

Start

R=7.1 m @

45


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Remember You can add vectors in any orderand yield

thesameresultant.


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ComponentsofVectors A=Ax +Ay Ax=A cos

Ay = A sin

Aslong as you

draw thexcomponentfirst

A

Ax

Ay


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Table Method Equation Add all X componentstogether FinalRx Add all Y componentstogether FinalRy


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Subtracting Vectors Simply add orsubtract 180 (keep

between 0 and 360) tothe

direction ofthe vectorbeingsubtracted (flip) You just ADD the OPPOSITE vector

(th

ereis

nosu

btraction in v

ectormath)


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Subtracting Vectors

vectors (physics)

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