Introduction To Vectors

Term Definition Notation Examples

il

Magnitude

-fhe amo\^ft* / s,ze o$r '-J*a" lua-ntt ly

i_j,IixI 5 E**tu

[* t,,?i

DirectionV.lh' nh L")ey {h. ve-*or

i * p oin*,r:g fu,"o,i

F*. -lfuo*r1,,;

f* 'o-

gl[t*r{

I^t

Lol

ScalarA nae. asure nre.d ftoonfi{y

t*:r*h n0 c\ire*iorr"" X = 5 ..,.,

t^iernr:ero.TurtlI

U rStqn c'LAqel'{q53

VectorA ft\ec$rr(e r\rr nT llun***;m'*h e dir cct-rcn

'}X= 5c"n

txtl

3r:plo c €. rr-rcrr tVe,{o ei *yForc e\rleiah*i-

UnitVector

A dec,fsr *ach+ i15

I un-{ long X= l*'nLNI

Equal

Vectors(Equivalent)

Vec*anE i-r".i rJ'l"r {{.e Ssnnerns a.,nr-rr, ce Grld

di rb"c*t {iv"-w

-) --.!.x -YTt i=5c"n[r

I I i=Ec*nr

Opposite

Vectors

Ve'ct*r* L.ulT l. rhiSn,r"re *oq" '** de *o*ofps*,*e *d,re**rcn*

*.>\x=-)-)X= 5 :"n (sJ

)= bcrnLt11

Parallel

Vectors

Ve***r* u;r?in -r*t g$crrng sr c:ilf#ri\ * e

di re *S or:":--> ,, -"5xl\ )

JX=bcmLN-:Y = 1 crnlhJtr/= lc"rftLTlJ

l

Coincident

Vectors

Veclnrs ro:i-t-h d'hqS*rrre dr recli*n "*{r*g-+

lie 0n *op #t eemh o*4rer

:-trl

l

Classifuing Vectors

For the following diagram, state:

a) Two vectors that are equal.

b) Two vectors that are opposite.

c) Two vectors that are coincident.

d) Two vectors that parallel, but have different magnitudes.

Drawing Vectors

. =5

FZ*3CD

\AF

-_-:"Ad**GH

\AK

\AI\

c6

d: 10 cm [N25oE]

Calculating Vectors

ualcrllate me maglutuoe ano crrecuon or tne rollowmg vectors:

N fVu""30:ut[*, 6',r't*E+ d , -z \a z*.'d"l + o-ic,u'+b'=c-* To.n€=9I i" (t+;ls) 'i 12 _/^ r_ r ? eI * '' /; rr"' t\-^' -Tin€= (t' ,,',, t:4,'

' iilaT_.:=

%n€=4' */*i- : qO = ct €i%ti-[31

Q{. ---: *

i ttrCx f €rgll.SJu(t,3)s, '-*{{\{lO =C

*)i*\*.'L^ * '.)\.a i \lJ - L\\u Unlt5 """-,

\ lstr\.iJo EJ_)

Calculate the magnitude and direction of the followigg vectors:

N ftho3"elf*, bLenS\h Farr^'^ia-

:iri - , j "'

' lc I =,Jfu2-xl * Cyr-y,)t

=.l(e;\L1y- \i?t* t+t

='tmI oflU = tv-

6-,--l /e,'

? e=lcn [f) €: 35,51"

2'5jb *c.

Lf 'qb,t-ll-U!

-lbnS = €

Vector Addition & Subtraction

Vector Addition

r To add vectors,line them up from h"o d rl-1- c=r L

r The sum of the vectors is drawn from *" il of &l. ,'",trr.l-L

to h{sd of lo;1.

r The sum of the vectors is also called the t-e: tu ! tu,:* ve c.to r

b>

' -l '.-nill ''r

z+O\

/- r' \g\tc(

V**y+QA+zq

\ \ ---.\ --}^J'PZuZ$.+iJjlr F:\L

-7&-

b)

'."i\f-

t''*---s\,q*11 )

State the resultant vector for each of the following:

a)

.\ F*

J

Determine i + u ir I ; 1= I.* 3[ r; F] .zcm, and rhe angle between rhe two vecrors is 40.00o.

l{et1'.n*_ t * MeqF}I,p /DfF /-I,VJ-f^3'-

Yl e fY'

tt''-k#

. -'-\i +'n

b

\-\*! i l , .

* i- ,,- i '.-! +. I uq ,n, ; a r'' -l't r'")%l

rillfltr "i p rr:i rn,:: e.*r

)r

,- " _-\-*:i

I

nn+ec+cD 1= AD )

b tt*

--.--*-----"

rYtu:',t r.B,,l

7

^*)o.

rr

F)trs*:- *ilt!it* I

* rfl*rf

*$ |Js'"

k3- t+On = g( zt. gb" ;[

1L

:,i rttf .T..r")

{}r: {: iJ'f i.{ T-{.

r so -rto {c f"rie, n)= !Lf0o

I)e>l

"t]t9.}

bfi

La,*;i("["L

t F-J -Flfi lC+ [1fI .,

-5+'*'o n*b= il.bJ.mt\+*--l\**-/\* **" ,..

- ,*.f ..

lr,I A4 rr An' r

Li 1" TU ri). .rr _'.

\rdl

Ari, i a i .F Ll{ll ";ohLcr,nleg_ -il..' :. --.' ,,. -

e xe-*)

Me"fi-ud1-/-!.r!-/-.

-. , !..,-.

L*J $

I i.,e iinnff PRCS

' eA ,'l f\i{ (r *)

, r f',;l.f e,, ,:rr'L'

C*ru-E*.* \*,.

' t {ar(E.l

..->--o- * b 2

t\,2tf. /, -? ..-{ i' wu u"-

i

"',--.J:a"

StnS='Ch

SrntS= \7.2

7"3"3'nt$=

x 5.52con

Co*S * -g-hCo*t{0 = i-a.feI)

?.2"tun*0 * K

</)*5 *, r

J.$-

l$"?3 -

n-j'2

o'+b"-- c-tC,-1t * '{. {n3e" f d' 6 I

/\lnrlu F -!-o-

+ .^ fl !+l^,-,H r -[*,LS rI \-Jl I \d

!-\ '-.f J

l/-.1" i r'i-r

-1: *r f 4"i,,':.;;; tilt, {*,'*-"\t0"7

13G "3512 : /i-B I

)l lr"6r -E;ETIF *r\i *- :

L23.3{r' }od-1 ,,

-. r.,..'_-_,.--._,,,.. i..- ., ' ..-.o'/

ho T)oe:F,r,ei #bl*-rrli"A"qle s t

- ttt,SBgq = - f 2l.g{etGs4Coa'/:rtL33Ei \

[-'rl'3{sK )=\*

lC.*?Z e ;.*.*

1? ?r ({- .lc J LS

1-\J\[*b )Vector Subtraction

r To subtract vectors, add the

. ..:d-u

<->--b

State the resultant vector for each of the following:

a) AB-CB-DC

of the vector being subtracted. (l I ; p )

b) w-LA- Aa - az

.'\-\ .,

Faltarw* re-: l. +*

i: -e-B = *.{o*,,* filstk d

/\! a L* ly-c tc t, A+ +" h*n* ? {.* "*eu. *.*}x {s rf;- t*

" Lo'*? r * ]. { { -:''.: il , l*-+'i i-.$r".#}v---1w\/q.fr\ ,4.2

V

r-i

3, !.1C * = - t+ 8. i-5 te Co.:S-t / 7" ?feLf s

I-s--'--#ir-

[-'tz .]';{t )-A /2v/

q 3"9q"* sZero Yector

x $ne

o A vector with a magnitude of

i.e. AB + BC - AC---.: .'*> ,\

--43*Bi + CA.

-*_\-AA---l:fi

and no rii rr.fi n ^ '

Vector Operations

Properties of Vectors Activity

Property #1

i+d=i

Wl False

Property fl4

;-6 = -6+i

@False

Property #7

ti-61+6 = i-G+i)r*@Property #10

3Qb=ZQi)-rfr-{ True } False\u-*J

Vector Operations

Property #5

;-6 = 6-;-'--\frrue Q$r)

Proporty #8

i+i=2i-/'-\{ qqg)7 ra'"

Property #11

zti+61 =2i+2i-rffit*r{ True)/ False*:--Y

Property #2

&-o=0

Property #3

i+6 = 6*i( True )/ False

Property 116

1a+6y+6=i+6+6)"ffir, *.( lrue /l false

Property ff9

i+6+i+6+i = 3i+26/-*ryttT./False

Properfy #12

2(;*b) =zi*zi

@Farse

@Fake

Simpli$ each of the following:

b)a) 3(5a+b) -(2b-aa).*\

\5d+36 -d { +dF l3S +T$

= \Bd -r zB: Od +CB

*-}

6Q; -26 + sdl. lth *t6t - rot:d - 6 + zil2'

*3otg$*R{oi*g*- ?04--". *ll'ruc

If a=2x+3y*4zand b=x+5z,express l0b*2a interrrsof x, y andz.

-5 -. -->lOb -')a-

Determine the valuel af l2i+306 | if 1 i 1: tO cnn, 6 is a unit vector, and the angle between the twovectors is 70o.

lzdl=

t30El

/:^izwf ,6!

,P .YrFl

Jo ;

t'16 rJtc

o4/2L'

e-- b +c

2Cro)2A c*-- 30*): $$ c r'^r

*-l8o--ts= ll00

(as.iq)

x( ro,2u)

- 2 bcCos AI

J t X *3cE\z =Jzo'*.3oLziziii:e:) cot I

t?d*3s61 ; ql"3lo **.,,*,t**),''

q *r;Y" '

?e.l t

Applications of Vectors - Velocity

r Velocity is a vector because it has both r^nn nrn ' trEtg una d,il***.*l # '*0'.:

\.J

. Air speed (water speed) is the speed of a plane (boat) relative to a person on board.

. Ground speed is the speed of a plane (boat) relative to a person on the ground and includes the

effect of wind (current).

Using vectors to represent velocities

Connie the canoeist wants to cross aive{'250m wide. The current flows at 5 kmih and Connie can

paddle at 13 km/h in still water. If she directs her boat towards a dock directly across the riverdetermine:

{ *' ''*'*"*- ' " 2€0 r.'.

a) her actual velocity as she crosses the river. I

) $ i,'*tf*

t3" *,3 n- [i It TinO = *

T,'r,A.}q= 4-' 7E'A

l-i* Tln Zt.0\ -X

Z ,Z 2?,tb=,)

qb. ll .-..' = \

rlHL+ \/

L-

TanS = 9-o* 1o_ 2t .O*

f"""**' -'*--* **"'-'*"1

'igL:s: "-€ao,rt"

r?.tliPn t 1-

a)

iI

)

fi-, rY j

,\-tl'Jlltlj}r'

I

ra t*rvrlFo

b)

-fan € =Ac{-

.,...-.,. -\, \.-/ ="1"-,"- ...

, \t...;he ps,li end *Ji ;

"'J$'n,^ *ri.

I

c) how long it takes her to cross the river.

(f . i-r-.J'r * l:t?, efui!'b-

\i i".

4) F?t t" ' t ''tl*'I lv'

fr.- - ---_ ."

,j]q1;". i !"'')l

-. -tu''n

'{

i* ^--** r r

1foil;1J t* LnnL:,1

25O"n3i.0..|"

fuAih,a, *t a)//6 I .(Lo rN

C"?1,18& kr^ z

A plane is steering at N30oW at an air speed (speed in still air) of 550 kmlh. If the wind is from

S50"W at 80 kmlh, find the ground speed and the course of the plane.

I crYl * lC l<'tvr.rMETHOD 1:

30-8- an d- l'/-

L#ffigD.*

Ha*4r"d J ;::-

L'Va4.*'"tr*

IvFFt@&3:

,-{rry- ' --'tn^' '' - '*-'-'-**""-\-.-

I -tv = 5?s H,,:li, lnr IL"WJt**'f"**;"-

^**,/\"*r&u*"""-,--*-.^n _-fqJ

t^,+ c- -Zbc CosA

fl 'S* ***e- t*r*slts's#; C" I r oC

,Q\{V/,

u)

50"

? t7"e-_b

ffi#{-

A^4ai b*+ ca- ?be (**A80"= 55f+ 5bl.3f1 ?(ssoXs6?.g-* Cob,

- 620282"11{19 = - (Zb?01Co* A

Coer{-rqorfrz.lqe1\ .n

l*r-uulnT l= n"

\

1.q3" ; A 30-1.15= 22-os

I

tsO -30-50= {00'

A plane is steering at N30"W at an air speed (speed in still air) of 550 km/h. The wind ir@ S50"Wat 80 km/h.

Find the ground speed and the course of the plane.

J*D-

Sin€ = o-h

Srn80 = 5

0'* f_ J503,89?+ 78tf . [i I 'fuArr-r?G= li I

5ro9,3T kmlh = ldl

Co:O = -LLCos 8C = -LtofffltupSS s x

l3.B? k*/[ iX

Z p.^*l-ernJ

5/"q 3\ k*'l tv

hr zr.osowlHow far will the plEne have travelled in 2h?

\d - v+

)

Tantj^ = 9-q,

Tan6, 78J85ffi"e1

€i='T,;,6' l7Flq\Fa3.rq

e = 1.15

-A-3Q - 1"15

'22,obo

J.'o V=

: 56q-39 xZ= lt38.lI krnr

-^{w' - ""v-"*-\."--}' =\-'-*-V-\*t "l i

{ ."-. The p\ane will haue *r*ve}led\o, I,'r=-.

I I SBJB kw, fr..r ag"#s* vfi )"'-"**"*A\'**----l^"-..'**,-/t"-*"--**t'*\*"***r*n\**-- 't''\*"^a'

8O Srn& =

Applications of Vectors - Force

o Force is something that either o. Pull S an object.

o Force is measured using the unit H.lgwlons fNJ .

o Amassoflkgexerrsaforce"r g"Bt N/k*, [do*da

Determine the downward force exerted by a 10 kg box of textbooks.

l0 kq r q.q-t_Nt

^r"*-rj=t,^.1 t

[J3*s;Components of Force

A force applied to an object can broken down into two components:

horizonfa-l

Verh c-o.l - rnove* ob\e-ct up /dowru

Marsha pushes shopping cart up a 15o incline using a force of 60 N. Calculate the horizontal and

vertical forces being exerted on the cart.

ftadry',a/

** -*?V'-o" *"'.*"-

!4er iamr"r*euX

\e.rh f;ffitr *

Sorr€ *;Sh

$un l%" f El-tun

tu#Sinl*"ldl*s.$3 N * lsl

r Sl.q kf S"SB hl

Cog lS. =

hl3L{sS

{-$rlC$sl$* lR!

Sl.g to r{ $ \RI

N ffi**m*dJ

o b're** $ot,.^,:ar

Cos€= 3*

l-rt{j

h

-dotaS1d

f"fx

Forces in Equilibrium

o When an object is .:.$*$-iCprl #. r" "'4: , an opposite equtJ force is

used to counteract the applied ror..r. F

-?eferrninq -t'lne -fgn5ioY) in o' ea.bl€- -#aq{ ie srrpp"rh'n3a- susgenA.d o.15 k3 pcndon-l li3h* S'x-lu"e .

II

I

II+

', a*

A piece of mobile aft is suspended from the ceiling using two cables that make angles of 65o and 70n

with the ceiling. If the mobile exerts a downward force of 48N, what is the tension in each cable?

lg0 -fog

' +50

{BO - rl5

't35'

I +tuV/

-.o The.l'il

I 80-1 0 --l oz 200

180-q0-tu5: t{$

r8C-lbs -pss- nnr$- {r#lJ

.-'-' ')t' -T-li ^'. I r\e

'"t!

'"'i'lq'r

.-,' .......,.,o;*'!""" "ir*n. "- "li':'''

:':'|s'i: ! nb.

.

4gr',nrs{*B t#'1 *{"'ry t q't*e * i}re ' *.,'J?,"'-l- "i t* f'tn:" 4': c"*rt,tot !

ig ' ruq " f *u" +* fl"sr h nHf " "T'

*'-A- "- '"t -'t'*"

ffi-jjr,?3-x 28, bq F.f

5*J3*' 4#e*L*ff ltrrlf,F*{$unl*S *Lffi Srn?

lfll = 1B$rn?a' -r 'Jrnr35

tensian in Y['"ir cetigJ" 3bJ5 t{ t*FJ--.-'' ... .-/'' ".. **,_J. ,_"."*_f

,/<:

IJ!1

Sin A -- 3io*B

0-b$r,1ffi' -?iGf+8 IRIj

In 15, n135* { ( ;: 'n?}

lH! - ih';$-f*r.fl#*.*,n B3S

W-rog--? s # l8o-: