2.1 Three Dimensional Curves and Surfaces


 Rosanna Preston
 2 years ago
 Views:
Transcription
1 . Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two or threedimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The line then is the line parallel to the vector v = (a, b, c) passing through the point P (,, ). In particular, if we view the vector as having its initial point at P, then we can move awa from P along the line b adding multiples of v. In other L.5. P,,.5. v a,b,c Figure : Parametric equation of a line. words, we can move tv along the line. In other words, the position of point Q(,, ) on the line is given b (,, ) = (,, ) + t(a, b, c) = ( + at, + bt, + ct). () This gives us the parametric equations of a line in 3space for a line passing through P (,, ) and parallel to v = ai + bj + ck : = + at, = + bt, = + ct. ()
2 Here t is the parameter that determines how far along the line ou have moved. Eample: Find the parametric equation of the line passing through P (, 3, ), parallel to i j + 3k. Solution: The line is given b parametric equations = + t, = 3 t, = + 3t. Note that these lines are infinite. If we want onl a line segment we must restrict the parameter. Thus, the line given in the eample can be restricted to the line segment joining P (, 3, ) to Q(3,, 5) if we restrict t to the interval t. Finall, two lines that are not parallel and do not cross are called skew lines.... Representing the parametric equation using vectors If instead of an initial point P (,, ), we define a position vector r = (,, ) to give an initial position on a line and r = (,, ) for an point on the line parallel to v = (a, b, c) through P, then equation () can be represented b the vector equation of a line as r = r + tv. (3) This is shown in Figure. Note that all these vectors are defined with initial points at the origin. This is entirel equivalent to the equations in (), but the notation is obviousl briefer, and the interpretation is purel in terms of vectors. If ou need to think about this more intuitivel, think of r as ordinar position vectors in space, r as a velocit vector and t as time. This then looks like one of the usual equations for linear motion in vector form. However, we can use this more generall for an parameter t and appropriate vector v. Let s return to the previous eample: Eample: Find the vector equation of the line passing through P (, 3, ), parallel to i j + 3k.. Solution: The initial position is given b the vector r = (, 3, ),
3 ..5 L P,..5 r r v t v Figure : Vector equation of a line r = r + vt. and the vector equation is r = (, 3, )+t(,, 3) = (+t, 3 t, +3t) = (+t)i+(3 t)j+(+3t)k. It is important to be comfortable with all these was of writing parametric equations of a line... Planes The first tpe of planes we might think of are the coordinate planes. The plane, for eample, is the set of a and with =. B etension, we can imagine planes parallel to the coordinate planes. For eample, the plane = a is the set of all and such that = a. These tpes of planes are shown in Figure 3. 3
4 a b c Figure 3: Planes parallel to the coordinate aes.... Planes specified b a point and normal vector An plane in 3space can be uniquel determined b giving a point on the plane and a vector perpendicular to the plane, called a normal vector. Suppose we want to find an equation of the plane passing through P (,, ) and perpendicular to n = (a, b, c). Let us define the vector r = (,, ) pointing to P, and r = (,, ) pointing to another point Q. Since n is perpendicular to the plane, n r = n r =, and therefore which in component form is written which results in the equation n (r r ) =, (4) (a, b, c) (,, ) =, (5) a( ) + b( ) + c( ) =. (6) This is known as the pointnormal form of the equation of a plane. Equation (4) is the vector form of this equation. Eample: Find the equation of the plane passing though (,, ) and with normal vector n = (, 3, ). Solution: We use equation (6) to give us ( ) + 3( ) + ( + ) =, 4
5 which can be simplified to =. In fact, this final form can be generalised to a + b + c + d =, (7) which is the equation with graph that is a plane with n = (a, b, c) as a normal. This is called the general form of the equation of a plane. d is determined b knowing a point that lies in the plane and substituting into the equation to find d. Other was to determine a plane are a point and two vectors parallel to the plane, or three points in the plane. Note that these allow us to find a normal vector and so the equation of the plane. The crossproduct of two vectors in the plane will give a vector normal to the plane. Three points P, P and P 3 allow us to define two vectors P P and P P 3, which again allow us to find the normal vector.... Intersecting planes Two distinct planes have an acute angle of intersection, θ π/. The angle is the same as either the angle between n and n or between n and n depending on the direction of the normal vectors. However, in either case the angle is given b cos θ = n n n n. (8) The absolute value ensures that regardless of the sign of the normal vectors we alwas have an acute angle. Eample: Find the angle between the planes + = 3, =. Solution: The normal vectors are given b n = (,, ) and n = (, 4, 4), and the angle is found from cos θ = = 8 (3)(6) = 4 9, 5
6 n n. Θ 5 Θ Figure 4: Intersecting planes. which gives the angle θ = cos 4 9 =.4 rad = 63.6o....3 Distance problems involving planes There are three distance problems we will be concerned with: the distance between a point and a plane; the distance between two parallel planes; find the distance between two skew lines. Theorem: The distance D between a point P (,, ) and the plane a + b + c + d = is D = a + b + c + d a + b + c. (9) Proof: Let Q(,, ) be a point in the plane, and n = (a, b, c) the normal vector with its initial point is at Q. Consider Figure 5. The distance D is equal to the length of the orthogonal projection of QP onto n. Therefore, recalling proj b v = v b b b, () 6
7 P,, proj n QP D Q,, Figure 5: Projection onto normal vector: distance from plane. we get However, we can write D = proj n QP = QP n n QP = (,, ) QP na( ) + b( ) + c( ), n = QP n. () n () and which together give us n = a + b + c, (3) D = a( ) + b( ) + c( ) a + b + c. (4) Moreover, since Q lies in the plane, it satisfies the equation a + b + c + d =, (5) which allows us to find d, d = a b c, (6) and therefore we find the result. 7
8 Eample: Find the distance between the point P (4, 4, ) and the plane =. Solution: Using () we get D = ()(4) + ( )(4) + (4)() = 6 6 =. (7) To compute the distance between two parallel planes, compute the distance between one plane and an point in the other plane. To find the distance between skew lines, define two parallel planes each of which contains one of the skew lines. Then the distance between the planes gives the distance between the skew lines. Eample: Find the distance between the skew lines L : = + 4t = 5 4t, = + 5t, L : = + 8t = 4 3t, = 5 + t. Solution: Let P and P be parallel planes containing L and L respectivel. We can find a point on each line and hence in each plane b setting t =, giving Q (, 5, ) and Q (, 4, 5), see Figure 6. Let s use Q and find the L Q D L Q Figure 6: Distance between skew lines. equation of the plane P. Since the planes are parallel, the vectors used to define the parametric equations of the lines u = (4, 4, 5) and u = (8, 3, ) are both parallel to P. Hence i j k n = u u = = i + 36j + k, 8 3 8
9 is normal to P and P. With this normal vector and the point Q, we find the equation of P : (c ) + 36( 4) + ( 5) =, which can be written in the general form =. Therefore the distance from P to the point Q (, 5, ) is D = ()() + (36)(5) + ()( ) = 95 87, which in turn is the distance from L to L, since the lie in the parallel planes...3 Quadric Surfaces The generalisation to the general quadratic equation?? which from which conic sections were derived in space is the seconddegree equation in, and, A + B + C + D + E + F + G + H + Ik + J =. (8) The graphs of this famil of equations are called the quadric surfaces. There are si common tpes of quadrics,. Ellipsoid:. Hperboloid of One Sheet: 3. Hperboloid of Two Sheets: a + b + c = a + b c = c a b = 4. Elliptic Cone: = a + b 5. Elliptic Paraboloid: = a + b (9) 6. Hperbolic Paraboloid: = b a where we assume that a, b, c >. These are shown in Figure 7. The have the following traces 9
10 . Ellipsoid: Traces in the coordinate planes are ellipses.. Hperboloid of One Sheet: The trace in the plane are ellipses, and traces in the  and planes are hperbolas. 3. Hperboloid of Two Sheets: There is no trace in the plane, although the traces in planes parallel to the plane are ellipses provided there is a trace, and traces in the  and planes are hperbolas. 4. Elliptic Cone: The trace in the plane is a point and in planes parallel to the plane the traces are ellipses, and traces in the  and planes are pairs of intersecting lines. 5. Elliptic Paraboloid: The trace in the plane is a point and in planes parallel to and above the plane the traces are ellipses, and traces in the  and planes are parabolas. 6. Hperbolic Paraboloid: The trace in the plane is a pair of intersecting lines, and traces in planes parallel to the plane are hperbolas, which open in the direction when above the plane and open in the  direction when below the plane. Traces in the  and planes are parabolas. These of course can also appear in other orientations along different coordinate aes, or indeed with crossproduct terms which would result in other orientations. If the elliptic crosssection of an elliptic cone or elliptic paraboloid is circular the are called a circular cone and a circular paraboloid respectivel. Of course an ellipsoid with all the crosssections circular is a sphere, i.e. a = b = c. You are not epected to accuratel draw an of these surfaces. If asked for a sketch, draw the traces on the planes, and join these to give a rough idea of the shape. As with the conic sections we can translate a quadric surface b moving awa from the origin to (a, b, c), which will result in the change (,, ) ( a, b, c) in the equations for the surfaces. However, this will not be required for this course.
11 Figure 7: Quadric surfaces.
12 ..4 Clindrical and Spherical Coordinates We have alread met polar coordinates in space. Now we introduce two coordinate sstems that are often useful when rectangular coordinates are awkward in 3space. Clindrical coordinates (ρ, θ, ): In terms of clindrical coordinates the rectangular coordinates are written = ρ cos θ, = ρ sin θ,, () where ρ <, θ < π. This is shown in Figure 8 If we wish to go Θ Ρ Figure 8: Clindrical coordinates (ρ, θ, φ). from rectangular to clindrical coordinates directl, we can use the relations ρ = +, tan θ =, =. () Spherical coordinates: In terms of spherical coordinates the rectangular coordinates are written = ρ sin φ cos θ, = ρ sin φ sin θ, = ρ cos φ, () where ρ <, θ < π and φ π. Figure 9 shows spherical coordinates. If we wish to go from rectangular to spherical coordinates directl,
13 Ρ Φ.5 Θ Figure 9: Spherical coordinates. we can use the relations ρ = + +, tan θ =, cos φ = + +. (3) Eample: Change + + = 9 to spherical polar coordinates. Solution: The coordinates for this surface (a sphere) are for ρ = 3 and so = 3 sin φ cos θ, = 3 sin φ sin θ, = 3 cos φ, (4) where θ and φ are the parameters. 3
Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155
Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate
More informationSection 11.4: Equations of Lines and Planes
Section 11.4: Equations of Lines and Planes Definition: The line containing the point ( 0, 0, 0 ) and parallel to the vector v = A, B, C has parametric equations = 0 + At, = 0 + Bt, = 0 + Ct, where t R
More informationMA261A Calculus III 2006 Fall Homework 3 Solutions Due 9/22/2006 8:00AM
MA6A Calculus III 6 Fall Homework Solutions Due 9//6 :AM 9. # Find the parametric euation and smmetric euation for the line of intersection of the planes + + z = and + z =. To write down a line euation,
More informationParametric Surfaces. Solution. There are several ways to parameterize this. Here are a few.
Parametric Surfaces 1. (a) Parameterie the elliptic paraboloid = 2 + 2 + 1. Sketch the grid curves defined b our parameteriation. Solution. There are several was to parameterie this. Here are a few. i.
More information12.5 Equations of Lines and Planes
Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P
More informationLecture 1 Introduction 1. 1.1 Rectangular Coordinate Systems... 1. 1.2 Vectors... 3. Lecture 2 Length, Dot Product, Cross Product 5. 2.1 Length...
CONTENTS i Contents Lecture Introduction. Rectangular Coordinate Sstems..................... Vectors.................................. 3 Lecture Length, Dot Product, Cross Product 5. Length...................................
More information42 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. Figure 1.18: Parabola y = 2x 2. 1.6.1 Brief review of Conic Sections
2 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.18: Parabola y = 2 1.6 Quadric Surfaces 1.6.1 Brief review of Conic Sections You may need to review conic sections for this to make more sense. You
More information1. x = 2 t, y = 1 3t, z = 3 2t. 4. x = 2+t, y = 1 3t, z = 3+2t. 5. x = 1 2t, y = 3 t, z = 2 3t. x = 1+2t, y = 3 t, z = 2+3t
Version 1 Homework 4 gri (11111) 1 This printout should have 1 questions. Multiplechoice questions ma continue on the net column or page find all choices before answering. CalC13e11b 1 1. points A line
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationMAT 1341: REVIEW II SANGHOON BAEK
MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and
More informationEquations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More informationLINEAR FUNCTIONS OF 2 VARIABLES
CHAPTER 4: LINEAR FUNCTIONS OF 2 VARIABLES 4.1 RATES OF CHANGES IN DIFFERENT DIRECTIONS From Precalculus, we know that is a linear function if the rate of change of the function is constant. I.e., for
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.114.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More informationDetermine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s
Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,
More informationSECTION 9.1 THREEDIMENSIONAL COORDINATE SYSTEMS 651. 1 x 2 y 2 z 2 4. 1 sx 2 y 2 z 2 2. xyplane. It is sketched in Figure 11.
SECTION 9.1 THREEDIMENSIONAL COORDINATE SYSTEMS 651 SOLUTION The inequalities 1 2 2 2 4 can be rewritten as 2 FIGURE 11 1 0 1 s 2 2 2 2 so the represent the points,, whose distance from the origin is
More informationSection 9.5: Equations of Lines and Planes
Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 35 odd, 237 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
More informationTwo vectors are equal if they have the same length and direction. They do not
Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must
More informationAx 2 Cy 2 Dx Ey F 0. Here we show that the general seconddegree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X
Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus
More informationL 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
More informationIdentifying second degree equations
Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +
More informationRotation of Axes 1. Rotation of Axes. At the beginning of Chapter 5 we stated that all equations of the form
Rotation of Axes 1 Rotation of Axes At the beginning of Chapter we stated that all equations of the form Ax + Bx + C + Dx + E + F =0 represented a conic section, which might possibl be degenerate. We saw
More information1.7 Cylindrical and Spherical Coordinates
56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a twodimensional coordinate system in which the
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationSection 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 2537, 40, 42, 44, 45, 47, 50
Section 11.1: Vectors in the Plane Page 779 Suggested Problems: 1, 5, 9, 17, 3, 537, 40, 4, 44, 45, 47, 50 Determine whether the following vectors a and b are perpendicular. 5) a = 6, 0, b = 0, 7 Recall
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationSection 2.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates
Section.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates The polar coordinate system consists of the origin O,the rotating ray or half line from O with unit tick. A point P in
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More informationSection 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables
The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,
More informationTriple Integrals in Cylindrical or Spherical Coordinates
Triple Integrals in Clindrical or Spherical Coordinates. Find the volume of the solid ball 2 + 2 + 2. Solution. Let be the ball. We know b #a of the worksheet Triple Integrals that the volume of is given
More information9.5 CALCULUS AND POLAR COORDINATES
smi9885_ch09b.qd 5/7/0 :5 PM Page 760 760 Chapter 9 Parametric Equations and Polar Coordinates 9.5 CALCULUS AND POLAR COORDINATES Now that we have introduced ou to polar coordinates and looked at a variet
More informationPhysics 53. Kinematics 2. Our nature consists in movement; absolute rest is death. Pascal
Phsics 53 Kinematics 2 Our nature consists in movement; absolute rest is death. Pascal Velocit and Acceleration in 3D We have defined the velocit and acceleration of a particle as the first and second
More informationSupporting Australian Mathematics Project. A guide for teachers Years 11 and 12. Algebra and coordinate geometry: Module 2. Coordinate geometry
1 Supporting Australian Mathematics Project 3 4 5 6 7 8 9 1 11 1 A guide for teachers Years 11 and 1 Algebra and coordinate geometr: Module Coordinate geometr Coordinate geometr A guide for teachers (Years
More information1. a. standard form of a parabola with. 2 b 1 2 horizontal axis of symmetry 2. x 2 y 2 r 2 o. standard form of an ellipse centered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
More informationREVIEW OF CONIC SECTIONS
REVIEW OF CONIC SECTIONS In this section we give geometric definitions of parabolas, ellipses, and hperbolas and derive their standard equations. The are called conic sections, or conics, because the result
More informationSection 2.4: Equations of Lines and Planes
Section.4: Equations of Lines and Planes An equation of three variable F (x, y, z) 0 is called an equation of a surface S if For instance, (x 1, y 1, z 1 ) S if and only if F (x 1, y 1, z 1 ) 0. x + y
More informationReview Sheet for Test 1
Review Sheet for Test 1 Math 26100 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And
More informationSection 13.5 Equations of Lines and Planes
Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines  specifically, tangent lines.
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More information10.5. Click here for answers. Click here for solutions. EQUATIONS OF LINES AND PLANES. 3x 4y 6z 9 4, 2, 5. x y z. z 2. x 2. y 1.
SECTION EQUATIONS OF LINES AND PLANES 1 EQUATIONS OF LINES AND PLANES A Click here for answers. S Click here for solutions. 1 Find a vector equation and parametric equations for the line passing through
More informationCOMPONENTS OF VECTORS
COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two
More informationSurface Normals and Tangent Planes
Surface Normals and Tangent Planes Normal and Tangent Planes to Level Surfaces Because the equation of a plane requires a point and a normal vector to the plane, nding the equation of a tangent plane to
More informationLines and Planes 1. x(t) = at + b y(t) = ct + d
1 Lines in the Plane Lines and Planes 1 Ever line of points L in R 2 can be epressed as the solution set for an equation of the form A + B = C. The equation is not unique for if we multipl both sides b
More information13.4 THE CROSS PRODUCT
710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationDr. Fritz Wilhelm, DVC,8/30/2004;4:25 PM E:\Excel files\ch 03 Vector calculations.doc Last printed 8/30/2004 4:25:00 PM
E:\Ecel files\ch 03 Vector calculations.doc Last printed 8/30/2004 4:25:00 PM Vector calculations 1 of 6 Vectors are ordered sequences of numbers. In three dimensions we write vectors in an of the following
More informationSection 7.2 Linear Programming: The Graphical Method
Section 7.2 Linear Programming: The Graphical Method Man problems in business, science, and economics involve finding the optimal value of a function (for instance, the maimum value of the profit function
More informationPractice Problems for Midterm 2
Practice Problems for Midterm () For each of the following, find and sketch the domain, find the range (unless otherwise indicated), and evaluate the function at the given point P : (a) f(x, y) = + 4 y,
More informationJUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson
JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3
More informationTriple integrals in Cartesian coordinates (Sect. 15.4) Review: Triple integrals in arbitrary domains.
Triple integrals in Cartesian coordinates (Sect. 5.4 Review: Triple integrals in arbitrar domains. s: Changing the order of integration. The average value of a function in a region in space. Triple integrals
More informationNotes on the representational possibilities of projective quadrics in four dimensions
bacso 2006/6/22 18:13 page 167 #1 4/1 (2006), 167 177 tmcs@inf.unideb.hu http://tmcs.math.klte.hu Notes on the representational possibilities of projective quadrics in four dimensions Sándor Bácsó and
More informationClick here for answers.
CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the ais and the tangent
More informationSection 105 Parametric Equations
88 0 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY. A hperbola with the following graph: (2, ) (0, 2) 6. A hperbola with the following graph: (, ) (2, 2) C In Problems 7 2, find the coordinates of an foci relative
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationSECTION 91 Conic Sections; Parabola
66 9 Additional Topics in Analtic Geometr Analtic geometr, a union of geometr and algebra, enables us to analze certain geometric concepts algebraicall and to interpret certain algebraic relationships
More informationSection 8.8. 1. The given line has equations. x = 3 + t(13 3) = 3 + 10t, y = 2 + t(3 + 2) = 2 + 5t, z = 7 + t( 8 7) = 7 15t.
. The given line has equations Section 8.8 x + t( ) + 0t, y + t( + ) + t, z 7 + t( 8 7) 7 t. The line meets the plane y 0 in the point (x, 0, z), where 0 + t, or t /. The corresponding values for x and
More informationIntegrals in cylindrical, spherical coordinates (Sect. 15.7)
Integrals in clindrical, spherical coordinates (Sect. 15.7) Integration in clindrical coordinates. eview: Polar coordinates in a plane. Clindrical coordinates in space. Triple integral in clindrical coordinates.
More informationthat satisfies (2). Then (3) ax 0 + by 0 + cz 0 = d.
Planes.nb 1 Plotting Planes in Mathematica Copright 199, 1997, 1 b James F. Hurle, Universit of Connecticut, Department of Mathematics, Unit 39, Storrs CT 66939. All rights reserved. This notebook discusses
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two nonzero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More information(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,
Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We
More informationCore Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More information1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.
.(6pts Find symmetric equations of the line L passing through the point (, 5, and perpendicular to the plane x + 3y z = 9. (a x = y + 5 3 = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
More informationGraphing Linear Equations
6.3 Graphing Linear Equations 6.3 OBJECTIVES 1. Graph a linear equation b plotting points 2. Graph a linear equation b the intercept method 3. Graph a linear equation b solving the equation for We are
More informationThe Graph of a Linear Equation
4.1 The Graph of a Linear Equation 4.1 OBJECTIVES 1. Find three ordered pairs for an equation in two variables 2. Graph a line from three points 3. Graph a line b the intercept method 4. Graph a line that
More informationTHE PARABOLA 13.2. section
698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More information5 VECTOR GEOMETRY. 5.0 Introduction. Objectives. Activity 1
5 VECTOR GEOMETRY Chapter 5 Vector Geometry Objectives After studying this chapter you should be able to find and use the vector equation of a straight line; be able to find the equation of a plane in
More information1.5 Equations of Lines and Planes in 3D
40 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3D Recall that given a point P = (a, b, c), one can draw a vector from
More informationCalculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
More information6.3 Parametric Equations and Motion
SECTION 6.3 Parametric Equations and Motion 475 What ou ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher... and wh
More informationChapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis
Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >
More informationSolving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Solving inequalities Jackie Nicholas Jacquie Hargreaves Janet Hunter c 6 Universit of Sdne Mathematics Learning Centre, Universit of Sdne Solving inequalities In these nots
More informationDouble Integrals in Polar Coordinates
Double Integrals in Polar Coordinates. A flat plate is in the shape of the region in the first quadrant ling between the circles + and +. The densit of the plate at point, is + kilograms per square meter
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review APPENDIX D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane Just as ou can represent real numbers b
More informationMath 259 Winter 2009. Recitation Handout 1: Finding Formulas for Parametric Curves
Math 259 Winter 2009 Recitation Handout 1: Finding Formulas for Parametric Curves 1. The diagram given below shows an ellipse in the plane. 51 13 (a) Find equations for (t) and (t) that will describe
More informationLines and Planes in R 3
.3 Lines and Planes in R 3 P. Daniger Lines in R 3 We wish to represent lines in R 3. Note that a line may be described in two different ways: By specifying two points on the line. By specifying one point
More informationCHAPTER 10 SYSTEMS, MATRICES, AND DETERMINANTS
CHAPTER 0 SYSTEMS, MATRICES, AND DETERMINANTS PRECALCULUS: A TEACHING TEXTBOOK Lesson 64 Solving Sstems In this chapter, we re going to focus on sstems of equations. As ou ma remember from algebra, sstems
More informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationM PROOF OF THE DIVERGENCE THEOREM AND STOKES THEOREM
68 Theor Supplement Section M M POOF OF THE DIEGENE THEOEM ND STOKES THEOEM In this section we give proofs of the Divergence Theorem Stokes Theorem using the definitions in artesian coordinates. Proof
More information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More informationREVIEW OF ANALYTIC GEOMETRY
REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.
More informationIn this this review we turn our attention to the square root function, the function defined by the equation. f(x) = x. (5.1)
Section 5.2 The Square Root 1 5.2 The Square Root In this this review we turn our attention to the square root function, the function defined b the equation f() =. (5.1) We can determine the domain and
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level  MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: 97 Mathematics
More informationMath 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t
Math 241 Lines and Planes (Solutions) The equations for planes P 1, P 2 and P are P 1 : x 2y + z = 7 P 2 : x 4y + 5z = 6 P : (x 5) 2(y 6) + (z 7) = 0 The equations for lines L 1, L 2, L, L 4 and L 5 are
More informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More information15.1. Exact Differential Equations. Exact FirstOrder Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact FirstOrder Equations 09 SECTION 5. Eact FirstOrder Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
More informationComplex Numbers. w = f(z) z. Examples
omple Numbers Geometrical Transformations in the omple Plane For functions of a real variable such as f( sin, g( 2 +2 etc ou are used to illustrating these geometricall, usuall on a cartesian graph. If
More informationWarmUp y. What type of triangle is formed by the points A(4,2), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D.
CST/CAHSEE: WarmUp Review: Grade What tpe of triangle is formed b the points A(4,), B(6, 1), and C( 1, 3)? A. right B. equilateral C. isosceles D. scalene Find the distance between the points (, 5) and
More informationVector Fields and Line Integrals
Vector Fields and Line Integrals 1. Match the following vector fields on R 2 with their plots. (a) F (, ), 1. Solution. An vector, 1 points up, and the onl plot that matches this is (III). (b) F (, ) 1,.
More informationAlgebra II: Strand 7. Conic Sections; Topic 1. Intersection of a Plane and a Cone; Task 7.1.2
1 TASK 7.1.2: THE CONE AND THE INTERSECTING PLANE Solutions 1. What is the equation of a cone in the 3dimensional coordinate system? x 2 + y 2 = z 2 2. Describe the different ways that a plane could intersect
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
More informationTo Be or Not To Be a Linear Equation: That Is the Question
To Be or Not To Be a Linear Equation: That Is the Question Linear Equation in Two Variables A linear equation in two variables is an equation that can be written in the form A + B C where A and B are not
More informationApplications of Trigonometry
5144_Demana_Ch06pp501566 01/11/06 9:31 PM Page 501 CHAPTER 6 Applications of Trigonometr 6.1 Vectors in the Plane 6. Dot Product of Vectors 6.3 Parametric Equations and Motion 6.4 Polar Coordinates 6.5
More informationJune 2011 PURDUE UNIVERSITY Study Guide for the Credit Exams in Single Variable Calculus (MA 165, 166)
June PURDUE UNIVERSITY Stud Guide for the Credit Eams in Single Variable Calculus (MA 65, 66) Eam and Eam cover respectivel the material in Purdue s courses MA 65 (MA 6) and MA 66 (MA 6). These are two
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More informationC1: Coordinate geometry of straight lines
B_Chap0_0805.qd 5/6/04 0:4 am Page 8 CHAPTER C: Coordinate geometr of straight lines Learning objectives After studing this chapter, ou should be able to: use the language of coordinate geometr find the
More informationJim Lambers MAT 169 Fall Semester 200910 Lecture 25 Notes
Jim Lambers MAT 169 Fall Semester 00910 Lecture 5 Notes These notes correspond to Section 10.5 in the text. Equations of Lines A line can be viewed, conceptually, as the set of all points in space that
More information