### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Week 3 MATH 210

Humboldt

GPA 3.7

### View Full Document

## 36

## 0

## Popular in Calculus 3

## Popular in Math

This 11 page Class Notes was uploaded by Merissa Notetaker on Sunday February 7, 2016. The Class Notes belongs to MATH 210 at Humboldt State University taught by Peter Goetz in Summer 2015. Since its upload, it has received 36 views. For similar materials see Calculus 3 in Math at Humboldt State University.

## Similar to MATH 210 at Humboldt

## Reviews for Week 3

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 02/07/16

Calculus 3 EX . Show that the equation ×2+y2+ -22-6×22=11 . 4y a sphere . Find center and radius if so . we around gives want move = ( X -a)2 + which is ' . / (X-3) + ( yt2)2t 1Z -1)2 . 25 a Sphere W/ center (3 , -2,1 ) radius 5 a square yes , c+2-2-22-+1 thy2t446×+9e+! ×2+bx+(b㱺 ?_ (b㱺2 3- Various Sub spaces of space (112) Coordinate Planes Z - : Z=O EX . Describe in words the set of points Xy plane equation 1123 8 and draw - . 1h that sahsty y< , ! XZ plane : equation y .o Z / y - plane : x.-o .8 plane in1123, line in1122 - yz equation y , × To , . R3={ (× ,y,z)/ × y ,z ER } ' ' parallel to xz plane 11 Containing ( 1,8 ,3) ⾨ Y × 8 YL is the set of all points in 1123 which he to the left of the .-8 . but not plane y on y¥ " pen half space shot lspace including boundary plane Cut in1/2 12.2 Vectors A vector is a that has a direction and an arrow representation quamty magnitude , using 11£11 Y Point Magnitude = length of arrow , 20 , µ,+,a| 1/22 i,to " . terminaa :}oint 1 X } 't'P j=u in the picture in general , T=h if and if 11Tltllhll and > have same direction only •X Describing Vector Algebra Addition to Tail Rule Tip Parallelogram Rule too % %t . :%t - T ! Math Fun Fact ! 1122 Friday Monty Hall Problem 3 game show doors class ywghoorIlln:e :#}ss , sw÷n switch doorsandor.on.si# .. what do you do ? ggg stay or switch . com 10 times switch time - 50% win loose Stay , play } every / Stats stay 66 .lu win for switch we a door W Chance we car. other doors has 213 chance : door dosh't chancek 1/3 got opening change of unpicked doors 12.2 vectors continued Zero vector 8 • 0+6 ¥ =L Scalar Multiplication r= some real number to make a new vector RT Case 2 : r< 0 : : easel r >o case 3 to , 08=8 rg " g. " oteillrvlrlrlnvn °€¥n9matghn,+ude=ryj magnitude = lryjy , ,, Subtraction of Vectors . A his F ::Oiit ... . * :# of Algebraic representation Vectors Z a 7 (a ,b,C) 8 in 3 . . µ space 0 to coordinate ze B we place tail at organ (0 ,0,01 y now it is defined as D= < a .b , (> X < Ex . Draw Vector I, -2,3 > , then move tail of Vector to ( Xiyiz ) Z ' ' ' 4-2/2-+3 > ( .pj= q<×+ / , - a( X -y, Z ) \ y x Vector From Coordinates - Given two points P= (X , ,, Z ,) Z. Q=( Xz , ,, Z .) , we get vector PQ , y y ,yz .-2 : 8 Xz - X,,,=( - y,, Z . -Z ,) o/Q=LXz yz X., ) Magnitude of Vector < a ,b, c) Unit Vector Coordinate Unit Vector Z ( a ,b,c) > W = < a, b ,c > is to 1 F< 1,010 > - magnitude equal I .-< o 1,0 ) . wy 11811 = ' 11611 : 1 , (o.O,o, NETC E=< ,) 0,0 , × y Vector yi. tt Unitizing in FE unit of I , making Vector in direction ÷ Proof : 11¥ 011 - " 811=1 th , ¥ Algebra wl coordmatized Vector li - L > it a ,b , c> J=< die ,f 8=( , ,bct+ef ) vii. ( ra > ,rb,rc Properties : Axioms of a vector Space . ' 1) uitv . Ttu 5) r( Itv )= rust rv litlw .it )= ( Prtw )tT 6) rts )X . ruitsu 2) ( 3) 8+8=8 7) ( vs )u>=r( sci ) 4) Ji ( ii)J=8 8) 1. ui=u 1125 Ex . > force P : Find resultant on F=t÷+F , ' = - > 11¥11 10lb F ,= 10 cos 45%+10 f- 552,552 ) F. 21 E s|n4S°j= o°j=( r F , 11FIH = 12lb Ez = 12 os 30 IZSM 60.3 6) .j⊥)g0 45 . ; ( .it } , P E , t¥z=( -552+653,552+6 )=E HE 11=5*6+1552.165-513.5 lb a- ' tan 'f¥2tt6ut)=76° 12.3 Dot Product . helps compute angles as = = La ,.dz , . ..an ) £ ( bi, bz , ... bn ) a%b=a . : scalar not a vector ,b tazbzt . . Ex . E. . ( 1,5 ) Tse ( - rz 3> Ex . as =P 6=25 . 3k 3 , .ij - space a=< 1,1 ,o> b- =< 0,2, 3> as .£=f| . - Feil ( 5.3 ) .' Bit 15 £ . } . 0.12+0=2 Properties r .tanbn real # any - 11 Ela ,, ... an> ai .ci . a ,2+ . .. tan = 11212 5) 8.2=0 as .B= 2) E.ci 3) as .( bit ) = oiob + as .E 4) ( rail . E=r( oiob ) Ex . Expression Does it make sense ? Result ( ciobloc no , cant dot scalar 3. Vector )E E ( oiib yes , you can scale vector Vector 11811 ( 5 ic ) scalar yes 8.5+2 no lloill (.tc ) no , cant dot scalar Is vector Angles Computi,g Pr00I : 0 < 0<180 ' Ther0= let 8 } } be nonzero vectors Law of cosine . Then T°T= 1151111811 COSO su .v )o ftp.ig#)hgC2=a2+b2-2abcosc • fyc- • } .ae O=co5' RH ' Huili11118?1 cos O ?_ ii. J.us -J - = iou us.T - 8 .b+ 8.8 utility 110412-26.81+11842 = . ) lid tlyvtlnylily flifolvlullllvllcoso - R.J. HRHHJIICOSO is Ex.as =iEtk, =35+25-K Finanglebtwn83.5 E.'b=6.2I 0=(054*5)×710 PerpendicuorOrthogonalectors two R-8 hf##E°J=o ias % 1126 EX. Parallelogram . .in "atano÷€¥#÷:II:IIId¥HtMI¥I¥¥¥on ,#¥ =( - Jot. tcuoultcuyvtvyuI ) EX.letP=(,-,-2, Q=(2,04, R=(,-,-5) = 2(io)i2(Tof) a) dotheyRakea triangle Parallel p. F¥:X¥¥ areheypara"e?' Fsanr:Xianiie 8 PR isnot ¥ Muttofsitlar aoscalmultipleFQ.p ie f=rP Yes.ita triangle b) Is DPQRa ? rightngleF• =0 acute 900 positivet 2 POTPT righbtwPQnandFR FQ.Frist3+6=1to obtuse90 negativet ( , ) =an.'eBQ. .÷f positive co( toll RP= -PR p: R%.RP=(-4,) °(5,-1,3= 20-2+3=21acute j: QDT=L-, -3,2.<,-,->= -4+6+-2=RigTriangle Directanglesnd Directionsines Ex. I=<,1,2 FindB. cos (u,T ) cosy co( a -c<iiiE)hdairsinisonseg )=11<2,1">= .it#.3 y ( i, D= co(s) 470.5 e=< to in any For Kn Project # I roggb=projectiononto =ra ¥÷ compaobillprogabllillbll,§j,F,ngiant proggb unitectordirection) ,gsj÷( iE÷h¥i=n:÷a=:÷a 57 from HW 3: 1/28 ,'s,㱺 gticDi : - Product and Work > force= F work = quanity.tt#es scalvectoristance vector a ¥ displacementa vector = FI Ellcosoo ND " Ex . E= 105+185 -6k is moving a box from point P=( 2,3,0 ) to point Q=( 4,9, 15) Assume units of force in N and distance in m . Find work D= Fa -< 2,6 157 , W= E .D= ( -6 ) . ( 2,6 15> = 20+108-90=38 NM = 38 J 10,18, , 12.4 Cross Product of Vectors Given two vectors in 3 . space , @ , b. Define oixh as a vector. oixb perpendicular to both in & 6 the page by > Ⱦ • Ex . a =< 1, 0,1> o 1,1> Find a vector E such that a × , . Two a ⊥ Z Ztb vectors \ E,a t,@ihst E=<×iYi㱺 choose az ycreates Eta . ,b=< ) Ⱦ E $ 2=1 , x=-1, y= I plane / Y .as=oȾ×+z=o .B=oȾy+z=o } SITE - x E+bȾe Ec .i, i.,> Definition of cixb Determinate of Matrices 2×2 matrix : ( ba ) Ⱦdet(a ba )= / bd|=ad= W Scalar Scalars 3×3 Mdtr " ; be , - bscz) - az( - - b. :( by big 9) , §÷agy)=|µ÷p+t§÷y|=%}( bsscntaslbkz of = (a Az as) D= b , , , < ,.bz , bz) &×b= - - - - Vector , aazbd}tlaibz ( aib} azb ) 'K} }÷ Md.k ,€ |=( : oixb Theron is perpendicular to 8×6 EX continued k=< 1,0, l), b=< o,1,1) axI=/&o ,of=1o - h()j+( to )k=( t , -1,1 >=E Which Vector , among all possible vectors perp . to oianbb , is oixb ? Vector : magnitude and direction •118611=1101111611 since µ ° 10 direction of oixb Is given by the € righthand rule ° >axB= - Bia > . 㱺 an Fact E is parallel tob a oixb O 1129 ! Fun Fact ! Friday Integer Brick Integer Table - l2tw2=a2 / w2+h2÷b2 NO One Knows wield :a" ' meta nt¥## A nat': wahraetantsooheurkonu l=4 D= 5 ' - W : W=3 , , } KEIHL : & wp€ was ,e=r integers ' msn integers Nporrornedasmaenrzone Lm=3 , n=z D= m -n ' } is Solutions ! n@ ng 12.4 Cross Product cont . l . oixb = ° IIQXB 11=1121111611 Sino • direction: hand rule o B To rector right Ex is perp to £} Torque : I > ilflrllrxfiliurnneusmo ⾨> E=i×÷ ¥ € ¥f and The Cross Product > Area Area of Triangles Area of parallogram = note: µ\ e- ztllaxbil h= 11611 Sino %s㱺|§ A=ualh=u£HH5y Sino = 11 oixbll Ex . P= ( 2,0,-3), Q= (3 , 1,0), R= ( 5,2 ,2) a) Find Vector perp to the plane determined points b) Find Area of by given Triangle .FI?IIkiIInxsiw..aeYIIEtt#ys.ea.%FEEEKiw.ztrs=Irz ± t# Volume and the Cross Product T - dot Pavr?Pedemlepl V= product ? TE llbxillheilbxcllllancoso =/ E .(BXEH ȼ absolute value, volume is positive! - h¥ ⾨ /Ⱦ area I Ex . E- (1,0 1> 5= (1,1, 37 E =( 3,2 57 , , , , - DxE=L .1, 4,1> at. ( 5×0=-1+0 1=-2 Ⱦ V= 1-21=2 2/1 Mount Ian Goat = ( )X ( mosquito ) Nothing ! scalar vector Cross Product Properties cix( btctaxhtaxc rlaxnitraxbiaxrsb ( itjlxcij ) . ixiiixjtjxi .jxj -3 . (-2 , / §+/=<÷ } , , ,, cixb = . Bia =L . 3 , 2,1 ) cixa -8 12.5 Lines and Planes Do lines in 3- have Z space slopes ? ZD ytmxtb Not really . its different . ° bLds or 2 points Slope is now Hewed as direction! !( Vector ) < • • s slope = rise no 0 (deahtotgso ! = anywhere IF I have a vector < aibic > does this determine a unique line? NO , but if we also have a point ! Line in 3- is determined : P=( X. , Z . and nonzero £=( aib c Space by yo, , ) : ^ tio Equation -(Xu ,y.,Z.) 1 Want Xiy ,Z in terms of Xo,yo ,-2. and a,b ,( of Yo t - some scalar : Can be time F . Q=( × ,z) × ,y F= d t.ir- LXa=Jp+FQOpted$ b = Xo > + at bt et ) ,y ,-27 < X.,yo,-2 . )ttLa , ,c> ( ,yo,Zo ( , , Xotat , yotbt , Zotct > Parametric Equations of a line X=Xo+ at Z=Zo+ct y=yo+bt Ex Find the vector and of line l ( -3 and . equation Parametric equations a through 1,0, ) (2/0,6) & P=Cl ,o,- 3) §=FQ= Ll, 0,9 ) < X ,y,-2 >=< 1,0, , 9) XZ por Q = ( 2,0, 6) plane % ×= , t , yWo e . × 2- plane Symmetric Equations oaib , call non Zero Ex . Find of a line parallel u solve for t to < -1ymmetric eq P= ) a , b5,C7 going through T ( 214,2 , , Xo Yo ,ZD YEEEO bEE¥=Z¥ t=x¥=¥y=ȼ¥ -2=2 parallel to ¥=y¥ y -z plane Find of . 2/2 Ex. parametric and symmetric eqns . a line through (2, 1,0 ? whose direction vector is perptojtttpt Xo=2 , yo =L , Zo=o Para: Symm : suppose 6=0 I=ci+is×cjx㱺= out ×¥' ' 2*9=1 -37++21,0of =t<.a.,b,c)tj=x<12-1 I> z = t ¥=y÷=÷ , , 2=-3+9 Planes # Point, Vector p = normal Vector( nonzero) Equation of a Plane 3=2+1 ^ 9 p (Xo Z ) rT=( aibic ) ambient % Th Itt P= , yo, . director plane T Ⱦ ¥ condition: FQLK FQ= . 0 ( scala) ,E¥X a 'ix±ifte¥:b a CZ = + Axtbyt axotbyo CZO =D axtbytcz P= : EX . let (3 ,-1,2 ) , Q=( 8,2 ,4 ), 12=1-1, -2,. 31 Find eqn of plane P ,Q3iR (Xo , , Zo) Containing • yo ri =FQxFR=s5 '' -5 p , 3,27×4-4 , ' .÷if=< . ,3 , > .#g '=det(§y & = , , • . La , b ,c ) : -13×+17 + 7z =L. 13)(-l) ȼ ( 17( 2)t ( 7)( -3) eq y = -42 infinite -13×-1 17yi7-2 1.1. answers ! Scalar multiples !! EX . Find of a -1 2,1 ) that contains the line of intersection of the : -Z =2 eqn plane through ( , planes ° oxty n¢ ZX - yt 3z= I - -2=2 . back in line . Xty Igvkalae z× , a 4) . 3 Plug Pointon \ 3×= - 5Ⱦ*z± . l÷ '¥' ' # \ ftp.32.IS -2=4 Esty u=z 41=0 #i€¥¥ykFrsec+•n #+y=u 4 we want 6 , we need in y= 6+32=23 another point plane line of intersections direction vector is perp to the normal s of two planes given vector to cross with at to find make D= (1,1 ,-1) x (2, -1,3 > rfdobfanptaonendh ⾨F' 12.5 Lines and Planes Continued z|y Between two - Angle Planes • in, If P , and Pz are planes in, acute ( 0<-0190 ) - > angle P = between A Az % - it > riz ¢ ( ,,Pz ) acute angle , } , q g- 0=2 where A , } Az are normal st P,,Pz if 5 ,Az)= cost this obtuse , ¢( , '(Ti°nI ) "e 790 ; just 11841115.11 subtract 180° Ex . p.: A,=< 1 1,1) Kina = ' +2+3=6 : , q= cos. , 6 = " .2° Pz Xtzyt Tz . < 1,2 ,3) , ( fsny ) ,,no,, Find (P ,P< 115<11--5147 ¢ , ) . to Draw two Find of lines of intersection of P ? Pz Try Planes parametric equations , : Z z P , . Pz" Pc : Nyt -2=1 Find 2 Find axis (o.O,) Pz: } !! intersection Xt2yt3Z=l points Found part ×+y+z=1 32=1 •0,0-+3 ) z to Ⱦ -×+ of plane = , • ⊥ ×+}[ / (hop• (91,0 Y do ,'zi0) Ⱦ y=T Xto +0=1 Ⱦ (l, 0,0) =p 0,0) Y Ⱦ ) -2 × ×=O ( - 2y+3z . / win plot graphed planes ! ! ¥=tȾ - 1=1 Ⱦ ( 0,2,.1) =Q y y=2Ⱦ PQ = d-. . - ( o -1,2 o, -1-0 >.< - 1,2, I> P= ( 1,0 .O) X=l -t , zt . t Could have , y , z= you crossed A , wl Az 12.6 and Quadnc Surface Cylinders unit Cylinder v We want an mb ," Circle in 2 - Z is axis of eq ( • -1 X2ty2=1 space 0 describing the • but in 3- space symmetry ( x,y ,z) that are 'M Of Z= ' ynyntnmefnts g×÷%The # ##¥ a cylinder : zdosnt appear - " ' z=t rulings vertical in this case X2tz2=1 EX EX . y=Z2 draw / describe the set of Points ( X,y,z ) satisfying eq Q$ Uzi ny 'z €€ 2- × ⾨T* space 3- space Fact Friday Fun 21g . Supertask Infinite amount of task in a finite amount of time. I 2 3 4 : 15 sec : task 3 After 2mm ? - Vsause - 5.6 in 3 out : youtube 314 in, 2 out In 30 sec : task 2 Ball I : task After task on T in. 21h , out 1mm 1 @ @ ... @ Graphing thing TEC . animation 12.6 Quadric Surface A surface is Set of all an the form quadnc the points ( X ,y ,Z ) that Satisfy equation of Ax 't - 2 A- J Bxytcxzt Dy2tEyz+Fz2+GxtHy+Iz+J=O degree polyn are Constants EX . spheres are quadric surface eqn of a sphere center ( a, bic) , radius r r- I X2+y 2+2-2=1 , . unit Sphere centered at Orgin (× - a )2+( - b)2+ ( Z -C )2=r2 Ⱦ foiled be 2 y out w/ poly quad : Ex . Z=x2+y2 gives a guadric surface . Compute and understand the traces ( X=n , y=n , Z n) °×=h Z=h2ty2 : trace parabola equation , ' n=lȾZ - : parabola at n .-1 Ity n=oȾz=yz n=II[€) y n= -1 Ⱦ z . y ⾨112+5=1+92 . * Is the same as x=n y=n y x z=n Ⱦ h=×2+y2 : n=2 n= 1 Ⱦ 1 = ×2ty2 circle n=o Ⱦ o= : C 0,0 ) circle th 1 -I . ×2+y2 • × n= - Ⱦ X2+y2 no complex numbers here ! n=0 - * L y x += ⾨ Zzx - Try +y2

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

#### "When you're taking detailed notes and trying to help everyone else out in the class, it really helps you learn and understand the material...plus I made $280 on my first study guide!"

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.