EQUATIONS OF LINES AND PLANES

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "EQUATIONS OF LINES AND PLANES"

Transcription

1 EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint form, symmetric equtions of line, degenerte cses where direction vector hs one or more coordinte zero, intersecting lines, eqution of plne, ngle etween plnes, line of intersection of plnes, distnce of point nd plne. Wht students should hopefully get: How the eqution setup reltes to the generl setup for curves nd surfces. Understnding of the degenerte cses. Role of prmeter restrictions in defining line segment. Deeper understnding of reltionship of direction vector nd direction cosines Direction cosines. Executive summry (1) For nonzero vector v, there re two unit vectors prllel to v, nmely v/ v nd v/ v. (2) The direction cosines of v re the coordintes of v/ v. if v/ v = l, m, n, then the direction cosines re l, m, nd n. We hve the reltion l 2 + m 2 + n 2 = 1. Further, if α, β, nd γ re the ngles mde y v with the positive x, y, nd z xes, then l = cos α, m = cos β, nd n = cos γ Lines. Words... (1) A line in R 3 hs dimension 1 nd codimension 2. A prmetric description of line thus requires 1 prmeter. A top-down equtionl description requires two equtions. (2) Given point with rdil vector r 0 nd direction vector v long line, the prmetric description of the line is given y r(t) = r 0 + tv. If r 0 = x 0, y 0, z 0 nd v =,, c, this is more explicitly descried s x = x 0 + t, y = y 0 + t, z = z 0 + tc. (3) Given two points with rdil vectors r 0 nd r 1, we otin vector eqution for the line s r(t) = tr 1 + (1 t)r 0. If we restrict t to the intervl [0, 1], then we get the line segment joining the points with these rdil vectors. (4) If the line is not prllel to ny of the coordinte plnes, this prmetric description cn e converted to symmetric equtions y eliminting the prmeter t. With the ove nottion, we get: = z z 0 c This is ctully two equtions rolled into one. (5) If c = 0 nd 0, the line is prllel to the xy-plne, nd we get the equtions:, z = z 0 Similrly for the other cses where precisely one coordinte is zero. (6) If = = 0 nd c 0, the line is prllel to the z-xis, nd we get the equtions: Actions... x = x 0, y = y 0 (1) To intersect two lines oth given prmetriclly: Choose different letters for prmeters, equte coordintes, solve 3 equtions in 2 vriles. Note: Expected dimension of solution spce is 2 3 = 1. 1

2 (2) To intersect line given prmetriclly nd line given y equtions: Plug in the coordintes s functions of prmeters into oth equtions, solve. Solve 2 equtions in 1 vrile. Note: Expected dimension of solution spce is 1 2 = 1. (3) To intersect two lines given y equtions: Comine equtions, solve 4 equtions in 3 vriles. Note: Expected dimension of solution spce is 3 4 = Plnes. Words... (1) Vector eqution of plne (for the rdil vector) is n (r r 0 ) = 0 where n is norml vector to the plne nd r 0 is the rdil vector of ny fixed point in the plne. This cn e rewritten s n r = n r 0. Using n =,, c, r = x, y, z, nd r 0 = x 0, y 0, z 0, we get the corresponding sclr eqution x + y + cz = x 0 + y 0 + cz 0. Set d = (x 0 + y 0 + cz 0 ) nd we get x + y + cz + d = 0. (2) The direction or prllel fmily of plne is determined y its norml vector. The ngle etween plnes is the ngle etween their norml vectors. Two plnes re prllel if their norml vecors re prllel. And so on. Actions... (1) Given three non-colliner points, we find the eqution of the unique plne contining them s follows: first we find norml vector y tking the cross product of two of the difference vectors. Then we use ny of the three points to clculte the dot product with the norml vector in the ove vector eqution or the corresponding sclr eqution. Note tht if the points re colliner, there is no unique plne through them ny plne contining their line is plne contining them. (2) We cn compute the ngle of intersection of two plnes y computing the ngle of intersection of their norml vectors. (3) The line of intersection of two plnes tht re not prllel cn e computed y simply tking the equtions for oth plnes. This, however, is not stndrd form for line in R 3. To find stndrd form, either find two points y inspection nd join them, or find one point y inspection nd nother point y tking the cross product of the norml vectors to the plne. (4) To intersect plne nd line, plug in prmetric expressions for the coordintes rising from the line into the eqution of the plne. We get one eqution in the one prmeter vrile. In generl, this is expected to hve unique solution for the prmeter. Plug in the vlue of the prmeter into the prmetric expressions for the line nd get the coordintes of the point of intersection. (5) For point with coordintes (x 1, y 1, z 1 ) nd plne x + y + cz + d = 0, the distnce of the point from the plne is given y x 1 + y 1 + cz 1 + d / c Lines nd plnes 1.1. Lines: dimension nd codimension. A line in R n hs dimension one nd codimension n 1. In prticulr, line in Eucliden spce R 3 hs dimension 1 nd codimension 3 1 = 2. In prticulr, sed on wht we know of dimension nd codimension, we expect tht: In top-down or reltionl description, we should need two independent equtions to define line. In ottom-up or prmetric description, we should need one prmeter to define line Plnes: dimension nd codimension. A plne in R 3 is 2-dimensionl, nd it hs codimension 3 2 = 1. In prticulr, sed on wht we know of dimension nd codimension, we expect tht: In top-down or reltionl description, we should need one eqution to define plne. In ottom-up or prmetric description, we should need two prmeters to define plne. This gets into the relm of functions of two vriles, so we will defer the ctul 2-prmeter description of plnes for now Intersection theory. We hve the following sic intersection fcts: Intersect Generic cse Specil cse 1 Specil cse 2 Specil cse 3 Plne, plne Line Empty (prllel plnes) Plne (equl plnes) Plne, line Point Empty (line prllel to, not on plne) Line (line on plne) Line, line Empty (skew lines) Point (intersecting lines) Empty (prllel lines) Line (equl lines) 2

3 The generic cse here represents the cse tht is most likely, i.e., the cse tht would rise if the things eing intersected were chosen rndomly. There re mthemticl wys of mking this precise, ut these re eyond the current scope. In prticulr, it is worth pointing out tht the generic cse is exctly s intersection theory predicts. Let s consider the three generic cses: Generic intersection of plne nd plne: A plne hs codimension 1, so the intersection of two plnes (genericlly) hs codimension 1+1 = 2. We know tht line hs codimension 2, so this mkes sense. Generic intersection of plne nd line: A plne hs codimension 1 nd line hs codimension 2, so the intersection of plne nd line (genericlly) hs codimension 1+2 = 3, so it is zero-dimensionl. A point is zero-dimensionl. Generic intersection of line nd line: A line hs codimension 2, so the intersection of two lines (genericlly) hs codimension = 4, so it hs dimension 3 4 = 1. Negtive dimension indictes tht the intersection is genericlly empty. After we study the intersection theory in detil for lines nd plnes, we will e in position to cquire etter understnding of the generl principles of intersection theory. Specificlly, we will cquire etter grsp of the non-generic cses where the intersections don t work out s they genericlly do. 2. Equtions of lines 2.1. The point-direction form. The generl principle ehind this is the sme s it is with the point-slope form. Bsiclly, to descrie line, it suffices to specify point on the line, nd the direction of the line. The direction is specified y specifying ny vector prllel to the line. Specificlly, given line with points A nd B on it, the direction of the line is given y tking the vector AB. Note tht ny two vectors tht re sclr multiples of ech other (i.e., prllel to ech other) specify the sme direction. Suppose r 0 is the rdil vector for one point on the line, nd v is ny nonzero vector long the line. Then the rdil vector (i.e., vector from the origin to point) for points on the line cn e defined y the prmetric eqution: r(t) = r 0 + tv where t vries over the rel numers. For ech vlue of t, we get rdil vector for some point on the line, nd every point on the line is covered this wy. Suppose r 0 = x 0, y 0, z 0 nd v =,, c. Then r 0 + tv is the vector: x 0 + t, y 0 + t, z 0 + tc The corresponding prmetric description of curve is: {(x 0 + t, y 0 + t, z 0 + tc) : t R} Note tht the choice of prmetric description depends on the choice of sepoint in the line nd the choice of vector (which cn e vried up to sclr multiples). By the wy, here is some terminology (which we overlooked erlier). The direction cosines for prticulr direction re defined s the coordintes of the unit vector in tht direction. The direction cosines of prticulr direction re denoted l, m, nd n. For instnce, if direction vector is 1, 2, 3, then the corresponding unit vector is 1/ 14, 2/ 14, 3/ 14, so the direction cosines re l = 1/ 14, m = 2/ 14, nd n = 3/ 14. The direction cosines re lso the cosines of the ngles mde y the vectors with the x-xis, y-xis, nd z-xis. They stisfy the reltion: l 2 + m 2 + n 2 = The two-point form. Suppose r 0 nd r 1 re the rdil vectors of two points on line. Then, we cn get line in the point-direction form y setting v = r 1 r 0. We thus get the form: This simplifies to: r(t) = r 0 + t(r 1 r 0 ) 3

4 r(t) = tr 1 + (1 t)r 0 As t vries over ll of R, this gives the whole line. When t = 0, we get the point with rdil vector r 0 nd when t = 1, we get the point with rdil vector r 1. If we llow only 0 t 1, we get the line segment joining the two points Top-down description: symmmetric equtions. To otin the symmetric equtions, we strt with the prmetric equtions nd then eliminte the prmeter. In other words, with the prmetric description: We note tht: {(x 0 + t, y 0 + t, z 0 + tc) : t R} x = x 0 + t, = t = Similrly, we get t = (y y 0 )/ nd t = (z z 0 )/c. Eliminting t, we get: = z z 0 c Note tht while this looks like single long eqution, it is ctully two equtions: nd y y 0 = z z 0 c This is in keeping with wht we expect/hope tht to descrie 1-dimensionl suset in 3-dimensionl spce, we need 3 1 = 2 equtions. Intuitively, wht these equtions re sying is tht the coordinte chnges re in the rtio : : c Exceptionl cse of lines prllel to one of the coordinte plnes. The symmetric equtions formultion reks down if one of the coordintes of the direction vector,, c is zero. In this cse, the line is prllel to one of the three coordinte plnes, with the third coordinte eing unchnged (e.g., if c = 0, then the line is prllel to the xy-plne, ecuse its z-coordinte is unchnged). They rek down even more when two coordintes of the direction vector re zero, which mens tht the line is prllel to one of the xes. In this cse, the symmetric equtions given ove do not work, nd we insted do the following. If only one coordinte of the direction vector is zero: If c = 0 nd, 0, then we get the two equtions:, z = z 0 Similrly for the other cses. If two coordintes re zero: If, sy = = 0, then we get the two equtions: x = x 0, y = y 0 z does not pper in the equtions ecuse it cn vry freely. This line is prllel to the z-xis. 4

5 2.5. Pirs of lines: questions out intersection. As we noted erlier, lines in R 3 hve codimension 2, so the intersection of two lines is expected to e empty. There re qulittively four possiilities: (1) The lines re skew lines: This is the most independent cse possile. Here, the equtions descriing the two lines re s independent of ech other s possile nd the two lines thus do not lie in the sme plne. They do not intersect. (2) The lines re intersecting lines in the sme plne: This is somewht less independent cse. Here, there is plne (not necessrily contining the origin) tht contins oth lines, nd the lines re not prllel, so they intersect t point. (3) The lines re prllel lines in the sme plne: Here, the equtions for the line re inconsistent in specific wy, so they lie in the sme plne ut re prllel. They do not intersect. Although the conclusion out intersection is the sme oth for pirs of prllel lines nd for pirs of skew lines, the resons ehind this conclusion re different. (4) The two lines re ctully the sme line: In this cse, their intersection is the sme line. This is the most dependent cse possile. We now exmine how to find the intersection of two lines. The pproch is simply specil cse of finding the intersection of two curves. Since the equtions re ll liner, we cn ctully devise specific procedures to solve the equtions. Both lines re given prmetriclly: In this cse, we first mke sure we hve different letters for the prmeters for ech line. Then we equte coordinte-wise nd solve the system of 3 liner equtions in 2 vriles (the prmeter vriles for the two lines). Note tht the numer of equtions is more thn the numer of vriles unsurprising since the generic cse is one of skew lines. After finding solutions for the two prmeters, plug ck to find the points. One line is given prmetriclly in terms of t, the other using symmetric equtions: We sustitute the prmetric expressions into the vlues of x, y, nd z in the symmetric equtions nd solve the system of two equtions in the one (prmeter) vrile t. After finding solutions for t, plug ck to find the points. Both lines re given y symmetric equtions: We solve ll the four symmetric equtions. 3. Plnes 3.1. Vector description in terms of dot product. For given plne in R 3, it either lredy psses through the origin, or there is unique plne prllel to it tht psses through the origin. We sy tht two plnes re prllel if they either coincide or they do not intersect equivlently, if for every line in one plne, there is line in the other plne prllel to it. A fmily of prllel plnes cn e thought of s shring direction. But how do we specify the direction of plne, which is two-dimensionl oject? The ide is to look t the complement, or the codimension, of the plne. Specificlly, we look t the direction tht is orthogonl to the plne. There is unique direction vector (up to sclr multiples) orthogonl to fmily of prllel plnes. Further, the dot product of this vector with the rdil vector in ny fixed plne in the fmily is constnt, nd this constnt differs for ech plne in the fmily. This llows us to give equtions for plnes s follows. Let n e norml vector (orthogonl vector) to plne nd let r 0 e the rdil vector for fixed point in the plne. Then, if r is the rdil vector for n ritrry point in the plne, we hve: Rerrnging, we get: n (r r 0 ) = 0 n r = n r 0 Note tht the right side is n ctul rel numer. If n =,, c nd r 0 = x 0, y 0, z 0, we get the sclr eqution: x + y + cz = x 0 + y 0 + cz 0 If we define d = (x 0 + y 0 + cz 0 ), we cn rewrite the ove s: 5

6 x + y + cz + d = 0 Conversely, ny eqution of the ove sort, where t lest one of the numers,, nd c is nonzero, gives plne Plne prllel to the coordinte xes nd plnes. We sy tht plne nd line re prllel if either the line lies on the plne or they do not intersect t ll. If = 0, the plne is prllel to the x-xis. If = 0, the plne is prllel to the y-xis. If c = 0, the plne is prllel to the z-xis. If = = 0, the plne is prllel to the xy-plne. If = c = 0, the plne is prllel to the yz-plne. If = c = 0, the plne is prllel to the xz-plne Finding the eqution of plne given three points. To specify plne, we need to provide t lest three points on the plne. Given these three points, we cn find the eqution of the plne s follows: We first tke two difference vectors nd tke their cross product to find norml vector to the plne: If the points re P, Q, nd R, we tke the difference vectors P Q nd P R nd compute their cross product. We now use the vector eqution, nd hence from tht the sclr eqution, tking ny of of the three points P, Q, or R s the sepoint. Note tht if the three points given re colliner, then they do not define unique plne. Rther, ny plne through the line joining these three points works. It is no surprise tht the ove procedure fils t the stge where we need to tke cross product, ecuse the cross product turns out to e the zero vector Intersecting two plnes: line of intersection. Given two plnes, the typicl cse is tht they intersect in line. If we hve sclr equtions for oth plnes, then the intersection line cn e descried y tking the two equtions together. Unfortuntely, this pir of two equtions together, while it does define line, is not directly one of the stndrd descriptions of line. There re mny wys of otining the line in stndrd form. One of these is s follows: first, find norml vectors to the plnes. For instnce, if the equtions for the plnes re: 1 x + 1 y + c 1 z + d 1 = 0 2 x + 2 y + c 2 z + d 2 = 0 Then the norml vectors to these plnes re 1, 1, c 1 nd 2, 2, c 2. A direction vector long the line of intersection must e perpendiculr to oth these norml vectors, hence, it must e in the line of the cross product. Hence, we tke the cross product 1, 1, c 1 2, 2, c 2. Now tht we ve found the direction vector long the intersection of these plnes, we need to find just one point long the intersection nd we cn then use the point-direction form. One wy of finding point is to set z = 0 in oth equtions nd solve the system for x nd y (this is ssuming tht neither is prllel to the xy-plne; otherwise choose some other coordinte). Note tht if the plnes re prllel or coincide, then their norml vectors re prllel nd thus the cross product of the norml vectors ecomes zero. Conversely, the cross product ecoming zero mens the plnes re prllel, so there is good reson for the line of intersection to not mke sense Intersecting two plnes: ngle of intersection. The ngle of intersection etween two plnes is the ngle of intersection etween their norml vectors. As for the line of intersection, we cn extrct the norml vector from the sclr eqution of the plnes. To compute the ngle of intersection, we use the formul s rc cosine of the quotient of the dot product y the product of the lengths. 6

7 3.6. Intersecting plne nd line. Given plne nd line, we cn intersect them s follows: If the plne is given y sclr eqution nd the line is given prmetriclly using prmeter t, then to compute the intersection, we plug in ll coordintes s functions of the prmeter into the sclr eqution for the plne, nd solve one eqution in the one vrile t. After finding the solution t, we plug this into the prmetric eqution of the line to find the coordintes of the point of intersection. There re three possiilities: The typicl cse is tht we hve liner eqution in one vrile, nd it hs unique solution. In other words, the plne nd line intersect t point. Another cse is tht the eqution simplifies to something nonsensicl, such s 0 = 1. In this cse, there is no intersection. Geometriclly, this mens the line is prllel to ut not on the plne. The finl cse is tht the eqution simplifies to tutology, such s 0 = 0. In this cse, ll rel t give solutions. Geometriclly, this mens tht the line is on the plne Distnce of point from plne. We will not hve much occsion to use this formul, ut we note it riefly nonetheless. Given point with coordintes (x 1, y 1, z 1 ) nd plne x + y + cz + d = 0, the distnce from the point to the plne is given y the formul: x 1 + y 1 + cz 1 + d c 2 7

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

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

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

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

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

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324 A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

More information

PHY 140A: Solid State Physics. Solution to Homework #2

PHY 140A: Solid State Physics. Solution to Homework #2 PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.

More information

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2 7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6

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

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a. Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles

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

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

Pure C4. Revision Notes

Pure C4. Revision Notes Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd

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

Summary: Vectors. This theorem is used to find any points (or position vectors) on a given line (direction vector). Two ways RT can be applied:

Summary: Vectors. This theorem is used to find any points (or position vectors) on a given line (direction vector). Two ways RT can be applied: Summ: Vectos ) Rtio Theoem (RT) This theoem is used to find n points (o position vectos) on given line (diection vecto). Two ws RT cn e pplied: Cse : If the point lies BETWEEN two known position vectos

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

Section 7-4 Translation of Axes

Section 7-4 Translation of Axes 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

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

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

Answer, Key Homework 8 David McIntyre 1

Answer, Key Homework 8 David McIntyre 1 Answer, Key Homework 8 Dvid McIntyre 1 This print-out should hve 17 questions, check tht it is complete. Multiple-choice questions my continue on the net column or pge: find ll choices before mking your

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

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

to the area of the region bounded by the graph of the function y = f(x), the x-axis y = 0 and two vertical lines x = a and x = b.

to the area of the region bounded by the graph of the function y = f(x), the x-axis y = 0 and two vertical lines x = a and x = b. 5.9 Are in rectngulr coordintes If f() on the intervl [; ], then the definite integrl f()d equls to the re of the region ounded the grph of the function = f(), the -is = nd two verticl lines = nd =. =

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

and thus, they are similar. If k = 3 then the Jordan form of both matrices is

and thus, they are similar. If k = 3 then the Jordan form of both matrices is Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

More information

Harvard College. Math 21a: Multivariable Calculus Formula and Theorem Review

Harvard College. Math 21a: Multivariable Calculus Formula and Theorem Review Hrvrd College Mth 21: Multivrible Clculus Formul nd Theorem Review Tommy McWillim, 13 tmcwillim@college.hrvrd.edu December 15, 2009 1 Contents Tble of Contents 4 9 Vectors nd the Geometry of Spce 5 9.1

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

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

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

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

4.11 Inner Product Spaces

4.11 Inner Product Spaces 314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

More information

Sirindhorn International Institute of Technology Thammasat University at Rangsit

Sirindhorn International Institute of Technology Thammasat University at Rangsit Sirindhorn Interntionl Institute of Technology Thmmst University t Rngsit School of Informtion, Computer nd Communiction Technology COURSE : ECS 204 Bsic Electricl Engineering L INSTRUCTOR : Asst. Prof.

More information

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

More information

Integration. 148 Chapter 7 Integration

Integration. 148 Chapter 7 Integration 48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

More information

Lesson 10. Parametric Curves

Lesson 10. Parametric Curves Return to List of Lessons Lesson 10. Prmetric Curves (A) Prmetric Curves If curve fils the Verticl Line Test, it cn t be expressed by function. In this cse you will encounter problem if you try to find

More information

Review guide for the final exam in Math 233

Review guide for the final exam in Math 233 Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered

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

A new algorithm for generating Pythagorean triples

A new algorithm for generating Pythagorean triples A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf

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

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001 CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic

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

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

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long

More information

Generalized Inverses: How to Invert a Non-Invertible Matrix

Generalized Inverses: How to Invert a Non-Invertible Matrix Generlized Inverses: How to Invert Non-Invertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax

More information

Regular Sets and Expressions

Regular Sets and Expressions Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite

More information

MODULE 3. 0, y = 0 for all y

MODULE 3. 0, y = 0 for all y Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)

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

Sequences and Series

Sequences and Series Centre for Eduction in Mthemtics nd Computing Euclid eworkshop # 5 Sequences nd Series c 014 UNIVERSITY OF WATERLOO While the vst mjority of Euclid questions in this topic re use formule for rithmetic

More information

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C; B-26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndom-numer genertor supplied s stndrd with ll computer systems Stn Kelly-Bootle,

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

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

Chapter 6 Solving equations

Chapter 6 Solving equations Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

Vector differentiation. Chapters 6, 7

Vector differentiation. Chapters 6, 7 Chpter 2 Vectors Courtesy NASA/JPL-Cltech Summry (see exmples in Hw 1, 2, 3) Circ 1900 A.D., J. Willird Gis invented useful comintion of mgnitude nd direction clled vectors nd their higher-dimensionl counterprts

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

Lecture 5. Inner Product

Lecture 5. Inner Product Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right

More information

Lecture 3 Basic Probability and Statistics

Lecture 3 Basic Probability and Statistics Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The

More information

Null Similar Curves with Variable Transformations in Minkowski 3-space

Null Similar Curves with Variable Transformations in Minkowski 3-space Null Similr Curves with Vrile Trnsformtions in Minkowski -spce Mehmet Önder Cell Byr University, Fculty of Science nd Arts, Deprtment of Mthemtics, Murdiye Cmpus, 45047 Murdiye, Mnis, Turkey. -mil: mehmet.onder@yr.edu.tr

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.

LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes. LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 64-83.

More information

CURVES ANDRÉ NEVES. that is, the curve α has finite length. v = p q p q. a i.e., the curve of smallest length connecting p to q is a straight line.

CURVES ANDRÉ NEVES. that is, the curve α has finite length. v = p q p q. a i.e., the curve of smallest length connecting p to q is a straight line. CURVES ANDRÉ NEVES 1. Problems (1) (Ex 1 of 1.3 of Do Crmo) Show tht the tngent line to the curve α(t) (3t, 3t 2, 2t 3 ) mkes constnt ngle with the line z x, y. (2) (Ex 6 of 1.3 of Do Crmo) Let α(t) (e

More information

Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.

Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep. Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:

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

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

Homework 10. Problems: 19.29, 19.63, 20.9, 20.68

Homework 10. Problems: 19.29, 19.63, 20.9, 20.68 Homework 0 Prolems: 9.29, 9.63, 20.9, 20.68 Prolem 9.29 An utomoile tire is inlted with ir originlly t 0 º nd norml tmospheric pressure. During the process, the ir is compressed to 28% o its originl volume

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL PRODUCTS AND FACTORIZATION MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

More information

r 2 F ds W = r 1 qe ds = q

r 2 F ds W = r 1 qe ds = q Chpter 4 The Electric Potentil 4.1 The Importnt Stuff 4.1.1 Electricl Potentil Energy A chrge q moving in constnt electric field E experiences force F = qe from tht field. Also, s we know from our study

More information

Uniform convergence and its consequences

Uniform convergence and its consequences Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,

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

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

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

More information

not to be republished NCERT POLYNOMIALS CHAPTER 2 (A) Main Concepts and Results (B) Multiple Choice Questions

not to be republished NCERT POLYNOMIALS CHAPTER 2 (A) Main Concepts and Results (B) Multiple Choice Questions POLYNOMIALS (A) Min Concepts nd Results Geometricl mening of zeroes of polynomil: The zeroes of polynomil p(x) re precisely the x-coordintes of the points where the grph of y = p(x) intersects the x-xis.

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

69. The Shortest Distance Between Skew Lines

69. The Shortest Distance Between Skew Lines 69. The Shortest Distnce Between Skew Lines Find the ngle nd distnce between two given skew lines. (Skew lines re non-prllel non-intersecting lines.) This importnt problem is usully encountered in one

More information

Variable Dry Run (for Python)

Variable Dry Run (for Python) Vrile Dr Run (for Pthon) Age group: Ailities ssumed: Time: Size of group: Focus Vriles Assignment Sequencing Progrmming 7 dult Ver simple progrmming, sic understnding of ssignment nd vriles 20-50 minutes

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

4.0 5-Minute Review: Rational Functions

4.0 5-Minute Review: Rational Functions mth 130 dy 4: working with limits 1 40 5-Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two

More information

Vectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics

Vectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics Chpter 2 Vectors nd dydics Summry Circ 1900 A.D., J. Willird Gis proposed the ide of vectors nd their higher-dimensionl counterprts dydics, tridics, ndpolydics. Vectors descrie three-dimensionl spce nd

More information

Brillouin Zones. Physics 3P41 Chris Wiebe

Brillouin Zones. Physics 3P41 Chris Wiebe Brillouin Zones Physics 3P41 Chris Wiebe Direct spce to reciprocl spce * = 2 i j πδ ij Rel (direct) spce Reciprocl spce Note: The rel spce nd reciprocl spce vectors re not necessrily in the sme direction

More information

In the following there are presented four different kinds of simulation games for a given Büchi automaton A = :

In the following there are presented four different kinds of simulation games for a given Büchi automaton A = : Simultion Gmes Motivtion There re t lest two distinct purposes for which it is useful to compute simultion reltionships etween the sttes of utomt. Firstly, with the use of simultion reltions it is possile

More information

Vectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics

Vectors and dyadics. Chapter 2. Summary. 2.1 Examples of scalars, vectors, and dyadics Chpter 2 Vectors nd dydics Summry Circ 1900 A.D., J. Willird Gis proposed the ide of vectors nd their higher-dimensionl counterprts dydics, tridics, ndpolydics. Vectors descrie three-dimensionl spce nd

More information

4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS

4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS 4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting

More information

Lectures 8 and 9 1 Rectangular waveguides

Lectures 8 and 9 1 Rectangular waveguides 1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves

More information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

More information

Double Integrals over General Regions

Double Integrals over General Regions Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing

More information

FAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University

FAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University SYSTEM FAULT AND Hrry G. Kwtny Deprtment of Mechnicl Engineering & Mechnics Drexel University OUTLINE SYSTEM RBD Definition RBDs nd Fult Trees System Structure Structure Functions Pths nd Cutsets Reliility

More information

Continuous Random Variables: Derived Distributions

Continuous Random Variables: Derived Distributions Continuous Rndom Vriles: Derived Distriutions Berlin Chen Deprtment o Computer Science & Inormtion Engineering Ntionl Tiwn Norml Universit Reerence: - D. P. Bertseks, J. N. Tsitsiklis, Introduction to

More information

Mathematics Higher Level

Mathematics Higher Level Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:

More information

Quadrilaterals Here are some examples using quadrilaterals

Quadrilaterals Here are some examples using quadrilaterals Qudrilterls Here re some exmples using qudrilterls Exmple 30: igonls of rhomus rhomus hs sides length nd one digonl length, wht is the length of the other digonl? 4 - Exmple 31: igonls of prllelogrm Given

More information

Quadratic Equations. Math 99 N1 Chapter 8

Quadratic Equations. Math 99 N1 Chapter 8 Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree

More information

A Note on Complement of Trapezoidal Fuzzy Numbers Using the α-cut Method

A Note on Complement of Trapezoidal Fuzzy Numbers Using the α-cut Method Interntionl Journl of Applictions of Fuzzy Sets nd Artificil Intelligence ISSN - Vol. - A Note on Complement of Trpezoidl Fuzzy Numers Using the α-cut Method D. Stephen Dingr K. Jivgn PG nd Reserch Deprtment

More information

Net Change and Displacement

Net Change and Displacement mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the

More information

Finite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh

Finite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 25 September 2015 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is

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

1.2 The Integers and Rational Numbers

1.2 The Integers and Rational Numbers .2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl

More information

Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra

Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra Sclr nd Vector Quntities : VECTO NLYSIS Vector lgebr sclr is quntit hving onl mgnitude (nd possibl phse). Emples: voltge, current, chrge, energ, temperture vector is quntit hving direction in ddition to

More information

Worksheet 24: Optimization

Worksheet 24: Optimization Worksheet 4: Optimiztion Russell Buehler b.r@berkeley.edu 1. Let P 100I I +I+4. For wht vlues of I is P mximum? P 100I I + I + 4 Tking the derivtive, www.xkcd.com P (I + I + 4)(100) 100I(I + 1) (I + I

More information