1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5.

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5."

Transcription

1 . Definition, Bsi onepts, Types. Addition nd Sutrtion of Mtries. Slr Multiplition. Assignment nd nswer key. Mtrix Multiplition. Assignment nd nswer key. Determinnt x x (digonl, minors, properties) summry 8. Inverse of x mtrix 9. Applitions: systems Crmer s Rule Prolem Solving Assignment nd nswer key 0. Prtie Exm nd Answer Key

2 Definition: A mtrix is retngulr rrngement of elements (entries) presented etween rkets [ ] or doule lines Exmples: [ ] g i

3 Bsi Conepts: Nming of mtrix: pitl letters re used to denote mtrix Mtrix Dimension: the numer of rows (horizontl) y the olumns (vertil) Given: 0 We n refer to this mtrix s: A x whih trnsltes s mtrix A hving dimensions y Note: dimension is lwys given s susript nd the vrile x reds y not times.

4 Types of mtries: ) row mtrix - ontins only one row [ ] ) olumn mtrix - ontins only one olumn ) zero mtrix - ll entries re zero d) trnspose mtrix - interhnging rows nd olumns A A T

5 Addition nd Sutrtion of Mtries Just s the sum or differene of two rel numers is unique rel numer, the sum or differene of two mtries is unique mtrix. 0 8 ( ( )) ( 0) ( 8) ( ( )) ( ( )) (( ) ) A B C x x x Only mtries of the sme dimensions n e dded or sutrted. If dimensions re different the ddition or sutrtion is undefined.

6 Mtrix ddition is oth ommuttive nd ssoitive: Given: 8 A, B C, Using the given informtion prove tht the given sttement is true. A) Commuttive A B B A or B C C B B) Assoitive A ( B C ) (A B) C

7 O mxn The sum of the zero mtrix nd ny other mtrix is A mxn, the zero mtrix is the identity element for ddition in the set of ny mtrix hving dimensions m x n A mxn Exmple: A mxn The dditive inverse (or negtive) of the mtrix is the mtrix - A mxn whose entries re the negtive of the orresponding entries in A. if A then A,, euse

8 As with rel numer, the onept of sutrtion of mtries n e redefined to s follows: the ddition of n dditive inverse A B A ( B ) mxn mxn mxn mxn 0

9 Slr Multiplition When we work with mtries we refer to ny rel numer s slr. The produt of slr s nd mtrix A mxn is the mtrix s A mxn eh of whose entries re s times the orresponding entry in A Exmple: 0 The produt of slr nd mtrix n e shown to e oth ommuttive nd ssoitive. This proof is left up to you.

10 Assignment: Given: A B [ ] C D E,,,, 9. Determine the dimensions of mtries A, B, nd C. Wht is the zero mtrix for mtrix B?. Wht is the trnspose of mtrix B? of mtrix E?. Wht is the dditive inverse for mtrix A? for mtrix D?. Determine the resultnt mtrix sed on the following opertions: ) x A or A ) - x C or -C ) (A D) E d) A -E e) D -(A E)

11 . Solve for the vrile mtrix: ) d 8 ) x y u v

12 Answer Key. A x, B x, C x....) B [ 0 0 0] B T, E T A D, 9 8.) ) d) e).) )

13 Mtrix Multiplition [ ] A x y z B Given: nd with the numers x, y, nd z representing the numer of ssettes, CD s nd DVD s sold eh week t lol retil outlet while the,, nd represent the prie of eh item during regulr sles week nd,, nd represent the prie of eh item during speil promotions week. Prolem: wht were the gross reeipts for the first weeks sles nd wht were they during the speil promotion. Note: Gross mount on one item numer of items x ost per item Gross reeipts totl of gross mounts for the vrious items Reeipts for st week

14 Gross reeipts for the first week ost of ssettes is x, of CD s y nd of DVD s z or x y z Gross reeipts for the speil promotion ost of ssettes is x, of CD s y nd of DVD s z or x y z The proess of dding the produts otined y multiplying the elements of row in one mtrix y the orresponding elements of olumn in nother mtrix defines the proess of mtrix multiplition

15 It is importnt to note tht for mtrix multiplition to our: ) the numer of elements in row in the first mtrix must mth the numer of elements in olumn in the seond mtrix ) the resulting nswer will lwys hve dimensions defined y the numer of rows in the first mtrix nd the numer of olumns in the seond mtrix Represent the dimensions of resulting mtrix A x * B x d C x d If equl, mtrix multiplition n our

16 Generl Notes: Mtrix multiplition differs from tht of rel numers in tht it is not, in generl, ommuttive. 0 nd Therefore, when it is neessry to find produt we must py strit ttention to the order in whih it is written. AB mens leftmultiplition of B y A, nd BA mens right-multiplition of B y A. if A nd B AB nd BA, 8 8

17 If we hve squre mtrix whose min digonl (from upper left to lower right) onsists of entries of I nd ll other entries re 0, we refer to this s n identity mtrix nd is leled I. The following re exmples of identity mtries for x nd x mtries nd Whether we use left or right multiplition, the identity mtrix times given mtrix results in the given mtrix

18 Given: illustrtes tht even though the produt of two mtries equls zero mtrix does not imply tht the first mtrix must equl zero or tht the seond mtrix must equl zero. If we were to ompre this with the rel numer system nd were given xy0, then either x 0 or y 0 Two vlid lws re left up to you to prove: Use the given mtries nd prove the following: ) The ssoitive lw (AB)C A(BC) ) The distriutive lws: AB AC A(B C) nd BACA (B C)A A B C,, 8

19 Assignment:... [ ] [ ]

20 Answers:..... [ ] [ ]

21 The Determinnt Funtion δ The ssoition of rel numer with squre mtries of ny dimension (order) is lled the determinnt of the mtrix. Determinnts n e distinguished from mtrix euse they re lwys enlosed within single set of vertil lines. The entries re lled elements of the determinnt nd the numer of entries in ny row or olumn is lled the order of the determinnt. The vlue of the rel numer defined y the rry is lulted y following definite rules referred to s expnding the determinnt.

22 To lulte the vlue of the determinnt A) For x determinnt: Given: A d Gret, formul tht mkes it esy Then: δ A d d Exmple #: B δ 8 B ( )( 8) ()( ) Exmple #: ( )( ) ( )()

23 Assignment: Clulte the vlue of eh x determinnt

24 Answer Key:

25 B. For x determinnt Proedure #- digonl method gol to disply sum of produts tht inludes ll possile rrngements of the susripts of the letters, nd. Given: Copy the first two olumns in order to the right of the rd olumn Multiply eh entry in first row y other two entries in the digonl moving from left to right (these produts re the first three terms of determinnt)

26 Multiply eh element in the row, strting from the right, y the other entries on the digonl going right to left. The negtives of these produts re the lst three terms of the determinnt ( ) ) ( δ Exmple: A A δ ( ) ( ) ( ) ) 0 ( ()()() )()() ( )()() ( )()() ( ()()() )()() ( A δ

27 Assignment: Clulte the vlue of eh determinnt using the digonl method.... 8

28 Answer Key:

29 Proedure # - Expnsion By Minors: A minor of n element in determinnt is the determinnt resulting from the deletion of the row nd olumn ontining the element. Given: The minor of in is The minor of in is

30 Steps:. Multiply eh element in the hosen row or olumn y its minor.. Determine whether eh element is to e ssigned or -. If the sum of the numer of the row nd the numer of the olumn ontining the element is odd ssign - nd if even ssign. Add the resulting produts Exmple: ( ) ( ) ( )( ) ( 0 ( )) () ( ( 8)) ( ) (9) ()

31 Assignment: Determine the vlue of eh determinnt using expnsion y minors. A. Aout indited row or olumn.. B. Aout ny row or olumn.. olumn ; 8 row ; 8 9 0

32 Answer Key:

33 Proedure #: Using properties of determinnts Property #: You n rete the negtive of n originl determinnt y interhnging ny two rows or olumns. - - nd 9-9 Property #: The determinnt hs vlue of 0 if two rows or olumns re identil Property#: The determinnt hs vlue of 0 if one row or olumn hs 0 for every element

34 Property #: We n rete determinnt equl to the originl if we interhnge ll the rows nd olumns in order. 0 nd 0 Property #: We n rete determinnt k times the originl if we multiply the elements of one row or olumn y rel numer k. 8 nd 8 8 Multiply st olumn y

35 And now for the most importnt property Property #: If eh element in ny row or olumn is multiplied y rel numer k nd the resulting produts re dded to the orresponding elements of nother row or olumn, the resulting determinnt equls the originl. 8 Multiply olumn y nd dd to (8) 8 8 olumn () 0 ( ) 9

36 Why is this so importnt?? This llows the mnipultion of the elements of the determinnt with the gol of mximizing the numer of elements equl to zero whih in turn llows expnsion y minors to fous on only one minor nd x determinnt. Exmple #: ( ) () () 8 Multiply nd row y nd dd to st row Multiply nd row y nd dd to rd row ( ) () () 0 9 Expnd y minors (8() (9)) 8 ( ) 9 0 8

37 Exmple #: x- () () ( ) () () () () () () x- x (( 0) ( )( ))

38 Assignment Clulte the vlue of the determinnt using the properties of determinnts nd reting zeros in rows nd olumns

39 Answer Key:

40 Summry out determinnts:. The formul d- works with n y x determinnt. The digonl method only works with x determinnts. Expnsion y minors will work with ny squre mtrix with dimensions three or greter.. The retion of zeros (row nd olumn opertions) provides the esiest pproh to determine the determinnt nd minimize the numer of lultion errors.

41 The Inverse of x Mtrix Rememer: results in the identity mtrix. When the produt of two rel numers is the multiplitive identity I, the two numers re lled multiplitive inverses. Similrly, ny two mtries A nd B suh tht AB BA I, re lled inverses. To identity the inverse of mtrix A, it is ustomry to sustitute A - for B How do we lulte the inverse??

42 Sure hope we hve formul!! Let u v A A AA I d nd x y suh tht u u u d x x dx v v y v 0 y dy For the ove sttement to e true: u u x v dx 0 v y dy 0

43 d 0 If, you n solve these pirs of equtions (sorry, this gret exerise is left up to you) for u nd x, nd for v nd y, respetively, nd find: u x d d d,, v y d d Sine eh denomintor is δ A, we n write A - s: A d δ A In Summry: To get A - ) interhnge the nd d ) hnge the signs on nd ) multiply the resulting mtrix y the determinnt of A.

44 Exmples: ) Determine the inverse mtrix for the given mtrix: ) Solve for mtrix A ) ()( ()() ; A A A A A

45 Assignment: A. Determine the inverse of the following mtries: B. Solve for the mtrix A... 8 A 9 A

46 Answer Key: 8 A B

47 Applitions of Mtries: ) Solving Systems of Equtions: y x y x y x Reson - mtrix multiplition If we re given: it is possile to write n equivlent mtrix eqution: nd y x y x y x Whih n e trnslted into: oeffiient mtrix X vrile mtrix onstnt mtrix

48 To solve: y x Rememer the use of the multiplitive inverse A y x y x δ A ) (, ) ( δ δ A y x ) ( ) ( δ δ Solution Set:

49 Exmple # Find the solution set for the given system: y x y x y x y x y x Mtrix eqution Multiplitive inverse Solution Set {(/9, -0/9)}

50 Exmple # Find the solution set for the following prolem. Use mtries. The mesure of one of two omplementry ngles is degrees less thn three times tht of the other. How lrge re the ngles? Let x mesure of lrger ngle in degrees y mesure of smller ngle in degrees Open sentenes: x y 90 x y - To use mtries it is required tht:. Both vriles must e on the left side of the equl sign nd the onstnt on the right side.tht the order of vriles e mintined in oth questions (use order sed on position in lphet)

51 90 90 y x y x y x Mtrix eqution Multiplitive Inverse Solution Set:{(, )}

52 B. Crmer s Rule Wht is this? A mtrix proedure tht n e used to solve systems of equtions - espeilly equtions tht hve three or more vriles. To understnd the rule let us first exmine the proedure s it would pply to system of equtions of two vriles y x y x nd Let x x xd D Coeffiient mtrix Property # y x y x xd Apply Property #

53 If D 0, x D D D x In summry:. Crete oeffiient mtrix D. Crete D x mtrix y repling the x oeffiients in the oeffiient mtrix with the onstnt vlues. The rtio D x /D defines the vlue for the vrile x.. Similrly, rete mtrix D Y y repling the y oeffiients in the oeffiient mtrix with the onstnt vlues. The rtio D y /D defines the vlue for the vrile y.. The replement of oeffiients with onstnts nd the retion of the rtio is repeted equl to the numer of different vriles

54 Exmple # Determine the solution set for the following system of equtions using Crmer s Rule x - y x y D D D x y ()() ()() ()() ( )() ( )() ()() x D D Dy, y D x Solution Set: {(/,-/)}

55 Exmple # Determine the solution set for the following system: (Rememer to lulte the vlue of the vrious determinnts you n use the digonl method or row or olumn opertions whih result in zeros) x - y z - -x y - z x y - z D 0 D x 8 Continues on next slide

56 0 z y D D 0 0, 0, 0 8 D D D D D D z y x Solution Set: {(9/0, /0, -/)}

57 C. Prolem Solving We wish to lulte the mximum height roket will reh, when will it reh this height, nd when it will hit the ground? A trking sttion ws le to give the ltitude of the roket t vrious horizontl distnes from the lunh site. The olleted dt ws s follows: ltitude of 8 kilometers t distne of kms, n ltitude of kms t 0 kms, nd n ltitude of 0 kms t kms. We will ssume tht the ojet in free flight will hve pth tht pproximtes prol defined y eqution f ( x) x x To use the eqution we will hve to determine the vlues of, nd y sustituting the oordintes of the positions we know we will rete three equtions.

58 f () f( 0) f () () ( 0) () () () 8 ( 0) or 0 or or To determine the vlues of, nd it is neessry to rete the D, D, D nd D determinnts D 00 D , D, D D D D D D D

59 The eqution of the pth of the ojet is: f ( x) 0.08x 8x A) to determine the horizontl distne t whih the roket rehes the mximum height x 8 ( 0.08) 0 B) to lulte the mximum height F (0 ) 0.08 (0 ) 8(0 ) mx height is 00 kilometers mximum height t 0 kms form lunh C) when the roket hits the ground (the ltitude of the roket will e 0) ± 8± 8 ( 0.08)(0) x ( 0.08) 8± nd 00 0 lunh position 00 point of impt

60 Assignment:. Solve the following system of equtions using the ide of multiplitive inverses. ) x - y ) 9x - y -x y -8 x y -. Solve the following systems of equtions using Crmer s Rule ) x -y - ) -x y z x - y x y - x - x y - z. A roket is trked t three distint points in flight. The position is reorded s pirs of the form (distne from lunh pd, ltitude). Find mximum height nd distne to impt from lunh site. (, 8), (, ), nd (, )

61 0. ) {(, )}. ) {(, )}, ) {( )} 9 9,, ) {( )}. Mximum height 0 distne to impt 0

62 Prtie exm: Given : ) A B C ) A B ) C T d) A*B C e) B - f) A..Solve: ) ) x y 9 -x y -, 8, C B A d δ

63 ) x y z - -x y z x y z -..Multiply the following: ) ) 8 0

64 .Determine the vlue of the determinnt using the indited method: ) digonl ) expnsion y minors ) properties of determinnts

65 Answers:. ) 0 ) ) d) 0 8 e) f) A) ) {(, )} ) {(,, )}. ) 8 ) 9 8. ) ) - ) -8

The area of the larger square is: IF it s a right triangle, THEN + =

The area of the larger square is: IF it s a right triangle, THEN + = 8.1 Pythgoren Theorem nd 2D Applitions The Pythgoren Theorem sttes tht IF tringle is right tringle, THEN the sum of the squres of the lengths of the legs equls the squre of the hypotenuse lengths. Tht

More information

Words Symbols Diagram. abcde. a + b + c + d + e

Words Symbols Diagram. abcde. a + b + c + d + e Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To

More information

The remaining two sides of the right triangle are called the legs of the right triangle.

The remaining two sides of the right triangle are called the legs of the right triangle. 10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right

More information

Right Triangle Trigonometry

Right Triangle Trigonometry CONDENSED LESSON 1.1 Right Tringle Trigonometr In this lesson ou will lern out the trigonometri rtios ssoited with right tringle use trigonometri rtios to find unknown side lengths in right tringle use

More information

excenters and excircles

excenters and excircles 21 onurrene IIi 2 lesson 21 exenters nd exirles In the first lesson on onurrene, we sw tht the isetors of the interior ngles of tringle onur t the inenter. If you did the exerise in the lst lesson deling

More information

ISTM206: Lecture 3 Class Notes

ISTM206: Lecture 3 Class Notes IST06: Leture 3 Clss otes ikhil Bo nd John Frik 9-9-05 Simple ethod. Outline Liner Progrmming so fr Stndrd Form Equlity Constrints Solutions, Etreme Points, nd Bses The Representtion Theorem Proof of the

More information

PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * challenge questions

PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * challenge questions PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * hllenge questions e The ll will strike the ground 1.0 s fter it is struk. Then v x = 20 m s 1 nd v y = 0 + (9.8 m s 2 )(1.0 s) = 9.8 m s 1 The speed

More information

8-1. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary

8-1. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary 8-1 The Pythgoren Theorem nd Its Converse Voulry Review 1. Write the squre nd the positive squre root of eh numer. Numer Squre Positive Squre Root 9 81 3 1 4 1 16 1 2 Voulry Builder leg (noun) leg Relted

More information

Thank you for participating in Teach It First!

Thank you for participating in Teach It First! Thnk you for prtiipting in Teh It First! This Teh It First Kit ontins Common Core Coh, Mthemtis teher lesson followed y the orresponding student lesson. We re onfident tht using this lesson will help you

More information

Lesson 18.2: Right Triangle Trigonometry

Lesson 18.2: Right Triangle Trigonometry Lesson 8.: Right Tringle Trigonometry lthough Trigonometry is used to solve mny prolems, historilly it ws first pplied to prolems tht involve right tringle. This n e extended to non-right tringles (hpter

More information

The AVL Tree Rotations Tutorial

The AVL Tree Rotations Tutorial The AVL Tree Rottions Tutoril By John Hrgrove Version 1.0.1, Updted Mr-22-2007 Astrt I wrote this doument in n effort to over wht I onsider to e drk re of the AVL Tree onept. When presented with the tsk

More information

Know the sum of angles at a point, on a straight line and in a triangle

Know the sum of angles at a point, on a straight line and in a triangle 2.1 ngle sums Know the sum of ngles t point, on stright line n in tringle Key wors ngle egree ngle sum n ngle is mesure of turn. ngles re usully mesure in egrees, or for short. ngles tht meet t point mke

More information

Chess and Mathematics

Chess and Mathematics Chess nd Mthemtis in UK Seondry Shools Dr Neill Cooper Hed of Further Mthemtis t Wilson s Shool Mnger of Shool Chess for the English Chess Federtion Mths in UK Shools KS (up to 7 yers) Numers: 5 + 7; x

More information

Ratio and Proportion

Ratio and Proportion Rtio nd Proportion Rtio: The onept of rtio ours frequently nd in wide vriety of wys For exmple: A newspper reports tht the rtio of Repulins to Demorts on ertin Congressionl ommittee is 3 to The student/fulty

More information

SOLVING EQUATIONS BY FACTORING

SOLVING EQUATIONS BY FACTORING 316 (5-60) Chpter 5 Exponents nd Polynomils 5.9 SOLVING EQUATIONS BY FACTORING In this setion The Zero Ftor Property Applitions helpful hint Note tht the zero ftor property is our seond exmple of getting

More information

In this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.

In this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists. Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix

More information

10.5 Graphing Quadratic Functions

10.5 Graphing Quadratic Functions 0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions

More information

THE PYTHAGOREAN THEOREM

THE PYTHAGOREAN THEOREM THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this

More information

SECTION 7-2 Law of Cosines

SECTION 7-2 Law of Cosines 516 7 Additionl Topis in Trigonometry h d sin s () tn h h d 50. Surveying. The lyout in the figure t right is used to determine n inessile height h when seline d in plne perpendiulr to h n e estlished

More information

D e c i m a l s DECIMALS.

D e c i m a l s DECIMALS. D e i m l s DECIMALS www.mthletis.om.u Deimls DECIMALS A deiml numer is sed on ple vlue. 214.84 hs 2 hundreds, 1 ten, 4 units, 8 tenths nd 4 hundredths. Sometimes different 'levels' of ple vlue re needed

More information

Simple Electric Circuits

Simple Electric Circuits Simple Eletri Ciruits Gol: To uild nd oserve the opertion of simple eletri iruits nd to lern mesurement methods for eletri urrent nd voltge using mmeters nd voltmeters. L Preprtion Eletri hrges move through

More information

Chapter. Contents: A Constructing decimal numbers

Chapter. Contents: A Constructing decimal numbers Chpter 9 Deimls Contents: A Construting deiml numers B Representing deiml numers C Deiml urreny D Using numer line E Ordering deimls F Rounding deiml numers G Converting deimls to frtions H Converting

More information

Practice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn

Practice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn Prtie Test 2 1. A highwy urve hs rdius of 0.14 km nd is unnked. A r weighing 12 kn goes round the urve t speed of 24 m/s without slipping. Wht is the mgnitude of the horizontl fore of the rod on the r?

More information

1. Area under a curve region bounded by the given function, vertical lines and the x axis.

1. Area under a curve region bounded by the given function, vertical lines and the x axis. Ares y Integrtion. Are uner urve region oune y the given funtion, vertil lines n the is.. Are uner urve region oune y the given funtion, horizontl lines n the y is.. Are etween urves efine y two given

More information

Basic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }

Basic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, } ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All

More information

Right Triangle Trigonometry 8.7

Right Triangle Trigonometry 8.7 304470_Bello_h08_se7_we 11/8/06 7:08 PM Pge R1 8.7 Right Tringle Trigonometry R1 8.7 Right Tringle Trigonometry T E G T I N G S T R T E D The origins of trigonometry, from the Greek trigonon (ngle) nd

More information

Lesson 2.1 Inductive Reasoning

Lesson 2.1 Inductive Reasoning Lesson.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 1, 16,,. 400, 00, 100, 0,,,. 1 8, 7, 1, 4,, 4.,,, 1, 1, 0,,. 60, 180, 10,

More information

Three squares with sides 3, 4, and 5 units are used to form the right triangle shown. In a right triangle, the sides have special names.

Three squares with sides 3, 4, and 5 units are used to form the right triangle shown. In a right triangle, the sides have special names. 1- The Pythgoren Theorem MAIN IDEA Find length using the Pythgoren Theorem. New Voulry leg hypotenuse Pythgoren Theorem Mth Online glenoe.om Extr Exmples Personl Tutor Self-Chek Quiz Three squres with

More information

DETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc.

DETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc. Chpter 4 DETERMINANTS 4 Overview To every squre mtrix A = [ ij ] of order n, we cn ssocite number (rel or complex) clled determinnt of the mtrix A, written s det A, where ij is the (i, j)th element of

More information

Solutions to Section 1

Solutions to Section 1 Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this

More information

Math Review for Algebra and Precalculus

Math Review for Algebra and Precalculus Copyrigt Jnury 00 y Stnley Oken. No prt of tis doument my e opied or reprodued in ny form wtsoever witout epress permission of te utor. Mt Review for Alger nd Prelulus Stnley Oken Deprtment of Mtemtis

More information

Proving the Pythagorean Theorem

Proving the Pythagorean Theorem Proving the Pythgoren Theorem Proposition 47 of Book I of Eulid s Elements is the most fmous of ll Eulid s propositions. Disovered long efore Eulid, the Pythgoren Theorem is known y every high shool geometry

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

11. PYTHAGORAS THEOREM

11. PYTHAGORAS THEOREM 11. PYTHAGORAS THEOREM 11-1 Along the Nile 2 11-2 Proofs of Pythgors theorem 3 11-3 Finding sides nd ngles 5 11-4 Semiirles 7 11-5 Surds 8 11-6 Chlking hndll ourt 9 11-7 Pythgors prolems 10 11-8 Designing

More information

State the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127

State the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127 ngles 2 CHTER 2.1 Tringles Drw tringle on pper nd lel its ngles, nd. Ter off its orners. Fit ngles, nd together. They mke stright line. This shows tht the ngles in this tringle dd up to 180 ut it is not

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

It may be helpful to review some right triangle trigonometry. Given the right triangle: C = 90º

It may be helpful to review some right triangle trigonometry. Given the right triangle: C = 90º Ryn Lenet Pge 1 Chemistry 511 Experiment: The Hydrogen Emission Spetrum Introdution When we view white light through diffrtion grting, we n see ll of the omponents of the visible spetr. (ROYGBIV) The diffrtion

More information

Homework Assignment 1 Solutions

Homework Assignment 1 Solutions Dept. of Mth. Sci., WPI MA 1034 Anlysis 4 Bogdn Doytchinov, Term D01 Homework Assignment 1 Solutions 1. Find n eqution of sphere tht hs center t the point (5, 3, 6) nd touches the yz-plne. Solution. The

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

Problem Set 2 Solutions

Problem Set 2 Solutions University of Cliforni, Berkeley Spring 2012 EE 42/100 Prof. A. Niknej Prolem Set 2 Solutions Plese note tht these re merely suggeste solutions. Mny of these prolems n e pprohe in ifferent wys. 1. In prolems

More information

8.2 Trigonometric Ratios

8.2 Trigonometric Ratios 8.2 Trigonometri Rtios Ojetives: G.SRT.6: Understnd tht y similrity, side rtios in right tringles re properties of the ngles in the tringle, leding to definitions of trigonometri rtios for ute ngles. For

More information

Final Exam covers: Homework 0 9, Activities 1 20 and GSP 1 6 with an emphasis on the material covered after the midterm exam.

Final Exam covers: Homework 0 9, Activities 1 20 and GSP 1 6 with an emphasis on the material covered after the midterm exam. MTH 494.594 / FINL EXM REVIEW Finl Exm overs: Homework 0 9, tivities 1 0 nd GSP 1 6 with n emphsis on the mteril overed fter the midterm exm. You my use oth sides of one 3 5 rd of notes on the exm onepts

More information

Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle

Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle : Using Trigonometry to Find Side Lengths of n Aute Tringle Clsswork Opening Exerise. Find the lengths of d nd e.. Find the lengths of x nd y. How is this different from prt ()? Exmple 1 A surveyor needs

More information

1 Fractions from an advanced point of view

1 Fractions from an advanced point of view 1 Frtions from n vne point of view We re going to stuy frtions from the viewpoint of moern lger, or strt lger. Our gol is to evelop eeper unerstning of wht n men. One onsequene of our eeper unerstning

More information

Quadratic Equations - 1

Quadratic Equations - 1 Alger Module A60 Qudrtic Equtions - 1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions - 1 Sttement of Prerequisite

More information

c b 5.00 10 5 N/m 2 (0.120 m 3 0.200 m 3 ), = 4.00 10 4 J. W total = W a b + W b c 2.00

c b 5.00 10 5 N/m 2 (0.120 m 3 0.200 m 3 ), = 4.00 10 4 J. W total = W a b + W b c 2.00 Chter 19, exmle rolems: (19.06) A gs undergoes two roesses. First: onstnt volume @ 0.200 m 3, isohori. Pressure inreses from 2.00 10 5 P to 5.00 10 5 P. Seond: Constnt ressure @ 5.00 10 5 P, isori. olume

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1. Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose

More information

How to Graphically Interpret the Complex Roots of a Quadratic Equation

How to Graphically Interpret the Complex Roots of a Quadratic Equation Universit of Nersk - Linoln DigitlCommons@Universit of Nersk - Linoln MAT Em Epositor Ppers Mth in the Middle Institute Prtnership 7-007 How to Grphill Interpret the Comple Roots of Qudrti Eqution Crmen

More information

Algebra Review. How well do you remember your algebra?

Algebra Review. How well do you remember your algebra? Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

More information

Right Triangle Trigonometry for College Algebra

Right Triangle Trigonometry for College Algebra Right Tringle Trigonometry for ollege Alger B A sin os A = = djent A = = tn A = = djent sin B = = djent os B = = tn B = = djent ontents I. Bkground nd Definitions (exerises on pges 3-4) II. The Trigonometri

More information

MATH PLACEMENT REVIEW GUIDE

MATH PLACEMENT REVIEW GUIDE MATH PLACEMENT REVIEW GUIDE This guie is intene s fous for your review efore tking the plement test. The questions presente here my not e on the plement test. Although si skills lultor is provie for your

More information

Angles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example

Angles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example 2.1 Angles Reognise lternte n orresponing ngles Key wors prllel lternte orresponing vertilly opposite Rememer, prllel lines re stright lines whih never meet or ross. The rrows show tht the lines re prllel

More information

Calculating Principal Strains using a Rectangular Strain Gage Rosette

Calculating Principal Strains using a Rectangular Strain Gage Rosette Clulting Prinipl Strins using Retngulr Strin Gge Rosette Strin gge rosettes re used often in engineering prtie to determine strin sttes t speifi points on struture. Figure illustrtes three ommonly used

More information

5.6 The Law of Cosines

5.6 The Law of Cosines 44 HPTER 5 nlyti Trigonometry 5.6 The Lw of osines Wht you ll lern out Deriving the Lw of osines Solving Tringles (SS, SSS) Tringle re nd Heron s Formul pplitions... nd why The Lw of osines is n importnt

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

GRADE 4. Fractions WORKSHEETS

GRADE 4. Fractions WORKSHEETS GRADE Frtions WORKSHEETS Types of frtions equivlent frtions This frtion wll shows frtions tht re equivlent. Equivlent frtions re frtions tht re the sme mount. How mny equivlent frtions n you fin? Lel eh

More information

SOLVING QUADRATIC EQUATIONS BY FACTORING

SOLVING QUADRATIC EQUATIONS BY FACTORING 6.6 Solving Qudrti Equtions y Ftoring (6 31) 307 In this setion The Zero Ftor Property Applitions 6.6 SOLVING QUADRATIC EQUATIONS BY FACTORING The tehniques of ftoring n e used to solve equtions involving

More information

The Pythagorean Theorem

The Pythagorean Theorem The Pythgoren Theorem Pythgors ws Greek mthemtiin nd philosopher, orn on the islnd of Smos (. 58 BC). He founded numer of shools, one in prtiulr in town in southern Itly lled Crotone, whose memers eventully

More information

Introduction. Law of Cosines. a 2 b2 c 2 2bc cos A. b2 a 2 c 2 2ac cos B. c 2 a 2 b2 2ab cos C. Example 1

Introduction. Law of Cosines. a 2 b2 c 2 2bc cos A. b2 a 2 c 2 2ac cos B. c 2 a 2 b2 2ab cos C. Example 1 3330_060.qxd 1/5/05 10:41 M Pge 439 Setion 6. 6. Lw of osines 439 Lw of osines Wht you should lern Use the Lw of osines to solve olique tringles (SSS or SS). Use the Lw of osines to model nd solve rel-life

More information

4.5 The Converse of the

4.5 The Converse of the Pge 1 of. The onverse of the Pythgoren Theorem Gol Use the onverse of Pythgoren Theorem. Use side lengths to lssify tringles. Key Words onverse p. 13 grdener n use the onverse of the Pythgoren Theorem

More information

Lecture 15 - Curve Fitting Techniques

Lecture 15 - Curve Fitting Techniques Lecture 15 - Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting - motivtion For root finding, we used given function to identify where it crossed zero where does fx

More information

The Pythagorean Theorem Tile Set

The Pythagorean Theorem Tile Set The Pythgoren Theorem Tile Set Guide & Ativities Creted y Drin Beigie Didx Edution 395 Min Street Rowley, MA 01969 www.didx.om DIDAX 201 #211503 1. Introdution The Pythgoren Theorem sttes tht in right

More information

Napoleon and Pythagoras with Geometry Expressions

Napoleon and Pythagoras with Geometry Expressions Npoleon nd Pythgors with eometry xpressions NPOLON N PYTORS WIT OMTRY XPRSSIONS... 1 INTROUTION... xmple 1: Npoleon s Theorem... 3 xmple : n unexpeted tringle from Pythgors-like digrm... 5 xmple 3: Penequilterl

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

Section 5-5 Solving Right Triangles*

Section 5-5 Solving Right Triangles* 5-5 Solving Right Tringles 379 79. Geometry. The re of retngulr n-sided polygon irumsried out irle of rdius is given y A n tn 80 n (A) Find A for n 8, n 00, n,000, nd n 0,000. Compute eh to five deiml

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

More information

Vectors Summary. Projection vector AC = ( Shortest distance from B to line A C D [OR = where m1. and m

Vectors Summary. Projection vector AC = ( Shortest distance from B to line A C D [OR = where m1. and m . Slr prout (ot prout): = osθ Vetors Summry Lws of ot prout: (i) = (ii) ( ) = = (iii) = (ngle etween two ientil vetors is egrees) (iv) = n re perpeniulr Applitions: (i) Projetion vetor: B Length of projetion

More information

Matrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA

Matrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA CHAPTER 1 Mtrix Algebr PREAMBLE Tody, the importnce of mtrix lgebr is of utmost importnce in the field of physics nd engineering in more thn one wy, wheres before 1925, the mtrices were rrely used by the

More information

Exponents base exponent power exponentiation

Exponents base exponent power exponentiation Exonents We hve seen counting s reeted successors ddition s reeted counting multiliction s reeted ddition so it is nturl to sk wht we would get by reeting multiliction. For exmle, suose we reetedly multily

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or more simply put, when adding or subtracting quantities, their uncertainties add. Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

More information

Chapter 9: Quadratic Equations

Chapter 9: Quadratic Equations Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

2.1 ANGLES AND THEIR MEASURE. y I

2.1 ANGLES AND THEIR MEASURE. y I .1 ANGLES AND THEIR MEASURE Given two interseting lines or line segments, the mount of rottion out the point of intersetion (the vertex) required to ring one into orrespondene with the other is lled the

More information

Solving Linear Equations - Formulas

Solving Linear Equations - Formulas 1. Solving Liner Equtions - Formuls Ojective: Solve liner formuls for given vrile. Solving formuls is much like solving generl liner equtions. The only difference is we will hve severl vriles in the prolem

More information

2 DIODE CLIPPING and CLAMPING CIRCUITS

2 DIODE CLIPPING and CLAMPING CIRCUITS 2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of

More information

LATTICE GEOMETRY, LATTICE VECTORS, AND RECIPROCAL VECTORS. Crystal basis: The structure of the crystal is determined by

LATTICE GEOMETRY, LATTICE VECTORS, AND RECIPROCAL VECTORS. Crystal basis: The structure of the crystal is determined by LATTICE GEOMETRY, LATTICE VECTORS, AND RECIPROCAL VECTORS Crystl bsis: The struture of the rystl is determined by Crystl Bsis (Point group) Lttie Geometry (Trnsltionl symmetry) Together, the point group

More information

The theorem of. Pythagoras. Opening problem

The theorem of. Pythagoras. Opening problem The theorem of 8 Pythgors ontents: Pythgors theorem [4.6] The onverse of Pythgors theorem [4.6] Prolem solving [4.6] D irle prolems [4.6, 4.7] E Three-dimensionl prolems [4.6] Opening prolem The Louvre

More information

Heron s Formula for Triangular Area

Heron s Formula for Triangular Area Heron s Formul for Tringulr Are y Christy Willims, Crystl Holom, nd Kyl Gifford Heron of Alexndri Physiist, mthemtiin, nd engineer Tught t the museum in Alexndri Interests were more prtil (mehnis, engineering,

More information

Geometry 7-1 Geometric Mean and the Pythagorean Theorem

Geometry 7-1 Geometric Mean and the Pythagorean Theorem Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the

More information

Sect 8.3 Triangles and Hexagons

Sect 8.3 Triangles and Hexagons 13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed two-dimensionl geometric figure consisting of t lest three line segments for its

More information

8. Hyperbolic triangles

8. Hyperbolic triangles 8. Hyperoli tringles Note: This yer, I m not doing this mteril, prt from Pythgors theorem, in the letures (nd, s suh, the reminder isn t exminle). I ve left the mteril s Leture 8 so tht (i) nyody interested

More information

Sine and Cosine Ratios. For each triangle, find (a) the length of the leg opposite lb and (b) the length of the leg adjacent to lb.

Sine and Cosine Ratios. For each triangle, find (a) the length of the leg opposite lb and (b) the length of the leg adjacent to lb. - Wht You ll ern o use sine nd osine to determine side lengths in tringles... nd Wh o use the sine rtio to estimte stronomil distnes indiretl, s in Emple Sine nd osine tios hek Skills You ll Need for Help

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

Chapter 10 Geometry: Angles, Triangles and Distance

Chapter 10 Geometry: Angles, Triangles and Distance hpter 10 Geometry: ngles, Tringles nd Distne In setion 1 we egin y gthering together fts out ngles nd tringles tht hve lredy een disussed in previous grdes. This time the ide is to se student understnding

More information

Right-angled triangles

Right-angled triangles 13 13A Pythgors theorem 13B Clulting trigonometri rtios 13C Finding n unknown side 13D Finding ngles 13E Angles of elevtion nd depression Right-ngled tringles Syllus referene Mesurement 4 Right-ngled tringles

More information

Exhaust System. Table of Contents. Section 6C - Measuring Exhaust Elbow Height

Exhaust System. Table of Contents. Section 6C - Measuring Exhaust Elbow Height Exhust System Mesuring Exhust Elow Height Tle of Contents Setion 6C - Mesuring Exhust Elow Height Mesuring Exhust Elow Height...6C-2 Generl Informtion...6C-2 Bot Requirements...6C-4 Loding Requirements...6C-5

More information

Square Roots Teacher Notes

Square Roots Teacher Notes Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this

More information

Boğaziçi University Department of Economics Spring 2016 EC 102 PRINCIPLES of MACROECONOMICS Problem Set 6 Answer Key

Boğaziçi University Department of Economics Spring 2016 EC 102 PRINCIPLES of MACROECONOMICS Problem Set 6 Answer Key Boğziçi University Deprtment of Eonomis Spring 2016 EC 102 PRINCIPLES of MACROECONOMICS Prolem Set 6 Answer Key 1. Any item tht people n use to trnsfer purhsing power from the present to the future is

More information

Activity I: Proving the Pythagorean Theorem (Grade Levels: 6-9)

Activity I: Proving the Pythagorean Theorem (Grade Levels: 6-9) tivity I: Proving the Pythgoren Theorem (Grde Levels: 6-9) Stndrds: Stndrd 7: Resoning nd Proof Ojetives: The Pythgoren theorem n e proven using severl different si figures. This tivity introdues student

More information

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right. Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction

More information

Lecture 2: Matrix Algebra. General

Lecture 2: Matrix Algebra. General Lecture 2: Mtrix Algebr Generl Definitions Algebric Opertions Vector Spces, Liner Independence nd Rnk of Mtrix Inverse Mtrix Liner Eqution Systems, the Inverse Mtrix nd Crmer s Rule Chrcteristic Roots

More information

MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!

MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent! MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

CONIC SECTIONS. Chapter 11

CONIC SECTIONS. Chapter 11 CONIC SECTIONS Chpter 11 11.1 Overview 11.1.1 Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig. 11.1). Fig. 11.1 Suppose we

More information