Suggested problems - solutions
|
|
- Bathsheba Hoover
- 6 years ago
- Views:
Transcription
1 Suggested problems - solutions Writing equations of lines and planes Some of these are similar to ones you have examples for... most of them aren t. P1: Write the general form of the equation of the plane containing the points (1, 3, 5), ( 2, 4, 1), and ( 1, 0, 1). Want: equation of a plane. Need: A point and a normal. Have: three points. Question becomes how to get normal. Relationship: Since the points are in the plane, we can write vectors v 1 and v 2 which lie in the plane as well. A normal to the plane is perpendicular to both vectors, so cross them. v 1 = AC =< 2, 3, 4 >. v 2 = AB =< 3, 1, 4 >. n = v 1 v 2 =< 16, 4, 11 >. Equation of plane given by n (r r 0 ) = 0, where r =< x, y, z >, and r 0 is position vector of any point on the plane; I ll use A for r 0 =< 1, 3, 5 >. < 16, 4, 11 > < x 1, y 3, z 5 > = 0 16(x 1) + 4(y 3) 11(z 5) = 0 16x y 12 11z + 55 = 0 16x + 4y 11z + 27 = 0 Final equation is in general form. You can check by verifying all three points satisfy the equation.
2 P2: Write the symmetric equations of a line through the point (1, 4, 1) and parallel to the line with symmetric equations x 2 = y = z 2 Want: equation of line parallel to given line. Need: a point and a direction vector for that line. Have: the point, and the given parallel line. Question becomes how to get direction vector for wanted line. Relationship: Direction vector for given L1 read off the equation. Since parallel, the lines are in the same direction: v 2 = v 2 =< 3, 4, 2 >. Point-parallel formula is < x, y, z >=< x 0, y 0, z 0 > +t < a, b, c >, so < x, y, z >=< 1, 4, 1 > +t < 3, 4, 2 > Since the problem requests final answer in symmetric form, solve it out: x = 1 + 3t t = x 1 3 y = 4 4t t = y 4 4 z = 1 2t t = z x 1 3 = y 4 4 = z + 1 2
3 P3: Write the point-normal form of the equation of a plane containing the line and the vector < 1, 1, 1 >. L : < x, y, z >=< 4, 1, 0 > +t < 2, 3, 4 > Want: equation of plane. Need: a point in the plane and a normal. Given: a line and a vector both in the plane. Relationship: The line is geometrically in the plane in space - any point on the line is a point on the plane. (You can t say this about vectors in a plane, by the way, because they represent the difference between points and move around in space.) So, any arbitary point on the line works as our point for the plane. Say (4, 1, 0). The direction vector for the line, v 1 =< 2, 3, 4 >, is in the plane. The other given vector v 2 =< 1, 1, 1 > is also in the plane. So the normal to the plane must be orthogonal to both; cross them. Point-normal equation: Stop there - that s the requested form. n = v 1 v 2 =< 7, 2, 5 > < 7, 2, 5 > < x 4, y 1, z 0 >= 0
4 P4: Write the point-parallel equation of a line through the point (1, 2, 1) and orthogonal to a plane containing the vectors < 1, 3, 2 > and < 10, 0, 1 >. Want: equation of a line. Need: point on line and direction vector for line. Have: vectors in a plane orthogonal to the line. Question becomes how to get direction vector for line - how is it related to normal for plane? Relationship: Since the line is orthogonal to the plane, it is parallel to the plane s normal. Direction vector v for the line is the same as normal n, and n can be found by crossing the vectors in the plane: Equation (point-parallel): v = n = v 1 v 2 =< 3, 21, 30 > < x, y, z >=< 1, 2, 1 > +t < 3, 21, 30 >
5 P5: Write the general form of the equation of the plane through the point (0, 0, 5) and parallel to the plane x y + 3z 6 = 0. Want: equation of plane. Need: point and normal. Have: point given. Have another parallel plane. How are normals related? Relationship: Parallel planes have parallel normals. n 2 = n 1 =< 1, 1, 3 >. Point-normal, solve to general: < 1, 1, 3 > < x 0, y 0, z 5 > = 0 1(x 0) 1(y 0) + 3(z 5) = 0 x y + 3z 15 = 0
6 P6: Write the general form of the equation of a plane through the point (0, 0, 5) and perpendicular to the plane x y + 3z 6 = 0. How many planes meet this description? Want: equation of plane. Need: point and normal. Have: point, and another perpendicular plane. How are normals related? Relationship: Since planes are orthogonal, so are the normals: n 1 n 2. That s not enough to fix a single direction; there are an infinite number of possibilities here. Since we need a vector orthogonal to only one given vector, we need a solution to n 1 n 2 = 0. Letting n 2 =< a, b, c >: < 1, 1, 3 > < a, b, c >= 0 a b + 3c = 0 Say a = 1, b = 2, so c = 1. The vector n 2 =< 1, 2, 1 > is orthogonal to the vector n 1 =< 1, 1, 3 >. Equation (solve to general): < 1, 2, 1 > < x 0, y 0, z 5 > = 0 x + 2y + z 5 = 0
7 P7: Find parametric equations for the line through the point (0, 1, 2) that is parallel to the plane x + y + z = 2 and orthogonal to the line x = 1 + t, y = 1 t, z = 2t. Want: equation of line. Need: point and direction vector. Have: point and parallel plane and orthogonal line. How is direction vector for line related to these two things? Relationship: v 2 v 1 (since lines are orthogonal). v 2 n (since lines parallel to plane, and thus orthogonal to normal). v 2 orthogonal to two known vectors, so cross them: Equation (solve to parametric): v 2 = n v 1 =< 1, 1, 1 > < 1, 1, 2 >=< 3, 1, 2 > < x, y, z >=< 0, 1, 2 > +t < 3, 1, 2 > x = 3t y = 1 t z = 2 2t
Review Exercise. 1. Determine vector and parametric equations of the plane that contains the
Review Exercise 1. Determine vector and parametric equations of the plane that contains the points A11, 2, 12, B12, 1, 12, and C13, 1, 42. 2. In question 1, there are a variety of different answers possible,
More information12.5 Lines and Planes in 3D Lines: We use parametric equations for 3D lines. Here s a 2D warm-up:
Closing Thu: 12.4(1)(2), 12.5(1) Closing next Tue: 12.5(2)(3), 12.6 Closing next Thu: 13.1, 13.2 12.5 Lines and Planes in 3D Lines: We use parametric equations for 3D lines. Here s a 2D warm-up: Consider
More informationSection 13.5: Equations of Lines and Planes. 1 Objectives. 2 Assignments. 3 Lecture Notes
Section 13.5: Equations of Lines and Planes 1 Objectives 1. Find vector, symmetric, or parametric equations for a line in space given two points on the line, given a point on the line and a vector parallel
More informationUpdated: January 11, 2016 Calculus III Section Math 232. Calculus III. Brian Veitch Fall 2015 Northern Illinois University
Math 232 Calculus III Brian Veitch Fall 2015 Northern Illinois University 12.5 Equations of Lines and Planes Definition 1: Vector Equation of a Line L Let L be a line in three-dimensional space. P (x,
More informationProblems of Plane analytic geometry
1) Consider the vectors u(16, 1) and v( 1, 1). Find out a vector w perpendicular (orthogonal) to v and verifies u w = 0. 2) Consider the vectors u( 6, p) and v(10, 2). Find out the value(s) of parameter
More informationRevision Problems for Examination 2 in Algebra 1
Centre for Mathematical Sciences Mathematics, Faculty of Science Revision Problems for Examination in Algebra. Let l be the line that passes through the point (5, 4, 4) and is at right angles to the plane
More informationVectors. Section 1: Lines and planes
Vectors Section 1: Lines and planes Notes and Examples These notes contain subsections on Reminder: notation and definitions Equation of a line The intersection of two lines Finding the equation of a plane
More informationMATH 200 (Fall 2016) Exam 1 Solutions (a) (10 points) Find an equation of the sphere with center ( 2, 1, 4).
MATH 00 (Fall 016) Exam 1 Solutions 1 1. (a) (10 points) Find an equation of the sphere with center (, 1, 4). (x ( )) + (y 1) + (z ( 4)) 3 (x + ) + (y 1) + (z + 4) 9 (b) (10 points) Find an equation of
More informationWriting Equations of Lines and Midpoint
Writing Equations of Lines and Midpoint MGSE9 12.G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel
More informationGEOMETRY IN THREE DIMENSIONS
1 CHAPTER 5. GEOMETRY IN THREE DIMENSIONS 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW GEOMETRY IN THREE DIMENSIONS Contents 1 Geometry in R 3 2 1.1 Lines...............................................
More informationDate Lesson TOPIC Homework. The Intersection of a Line with a Plane and the Intersection of Two Lines
UNIT 4 - RELATIONSHIPS BETWEEN LINES AND PLANES Date Lesson TOPIC Homework Oct. 4. 9. The Intersection of a Line with a Plane and the Intersection of Two Lines Pg. 496 # (4, 5)b, 7, 8b, 9bd, Oct. 6 4.
More information.(3, 2) Co-ordinate Geometry Co-ordinates. Every point has two co-ordinates. Plot the following points on the plane. A (4, 1) D (2, 5) G (6, 3)
Co-ordinate Geometry Co-ordinates Every point has two co-ordinates. (3, 2) x co-ordinate y co-ordinate Plot the following points on the plane..(3, 2) A (4, 1) D (2, 5) G (6, 3) B (3, 3) E ( 4, 4) H (6,
More informationSection 8.3 Vector, Parametric, and Symmetric Equations of a Line in
Section 8.3 Vector, Parametric, and Symmetric Equations of a Line in R 3 In Section 8.1, we discussed vector and parametric equations of a line in. In this section, we will continue our discussion, but,
More informationThe Three Dimensional Coordinate System
The Three-Dimensional Coordinate System The Three Dimensional Coordinate System You can construct a three-dimensional coordinate system by passing a z-axis perpendicular to both the x- and y-axes at the
More informationPre-Calculus Guided Notes: Chapter 10 Conics. A circle is
Name: Pre-Calculus Guided Notes: Chapter 10 Conics Section Circles A circle is _ Example 1 Write an equation for the circle with center (3, ) and radius 5. To do this, we ll need the x1 y y1 distance formula:
More informationGeometry Pre AP Graphing Linear Equations
Geometry Pre AP Graphing Linear Equations Name Date Period Find the x- and y-intercepts and slope of each equation. 1. y = -x 2. x + 3y = 6 3. x = 2 4. y = 0 5. y = 2x - 9 6. 18x 42 y = 210 Graph each
More informationSlope, Distance, Midpoint
Line segments in a coordinate plane can be analyzed by finding various characteristics of the line including slope, length, and midpoint. These values are useful in applications and coordinate proofs.
More informationAlgebra 1 Semester 2 Final Review
Team Awesome 011 Name: Date: Period: Algebra 1 Semester Final Review 1. Given y mx b what does m represent? What does b represent?. What axis is generally used for x?. What axis is generally used for y?
More informationSection Graphs and Lines
Section 1.1 - Graphs and Lines The first chapter of this text is a review of College Algebra skills that you will need as you move through the course. This is a review, so you should have some familiarity
More information12.7 Tangent Planes and Normal Lines
.7 Tangent Planes and Normal Lines Tangent Plane and Normal Line to a Surface Suppose we have a surface S generated by z f(x,y). We can represent it as f(x,y)-z 0 or F(x,y,z) 0 if we wish. Hence we can
More informationTopic 1.6: Lines and Planes
Math 275 Notes (Ultman) Topic 1.6: Lines and Planes Textbook Section: 12.5 From the Toolbox (what you need from previous classes): Plotting points, sketching vectors. Be able to find the component form
More informationChapter 1. Linear Equations and Straight Lines. 2 of 71. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 1 Linear Equations and Straight Lines 2 of 71 Outline 1.1 Coordinate Systems and Graphs 1.4 The Slope of a Straight Line 1.3 The Intersection Point of a Pair of Lines 1.2 Linear Inequalities 1.5
More information1.8 Coordinate Geometry. Copyright Cengage Learning. All rights reserved.
1.8 Coordinate Geometry Copyright Cengage Learning. All rights reserved. Objectives The Coordinate Plane The Distance and Midpoint Formulas Graphs of Equations in Two Variables Intercepts Circles Symmetry
More informationThe figures below are all prisms. The bases of these prisms are shaded, and the height (altitude) of each prism marked by a dashed line:
Prisms Most of the solids you ll see on the Math IIC test are prisms or variations on prisms. A prism is defined as a geometric solid with two congruent bases that lie in parallel planes. You can create
More informationLines and Planes in 3D
Lines and Planes in 3D Philippe B. Laval KSU January 28, 2013 Philippe B. Laval (KSU) Lines and Planes in 3D January 28, 2013 1 / 20 Introduction Recall that given a point P = (a, b, c), one can draw a
More information3, 10,( 2, 4) Name. CP Algebra II Midterm Review Packet Unit 1: Linear Equations and Inequalities. Solve each equation. 3.
Name CP Algebra II Midterm Review Packet 018-019 Unit 1: Linear Equations and Inequalities Solve each equation. 1. x. x 4( x 5) 6x. 8x 5(x 1) 5 4. ( k ) k 4 5. x 4 x 6 6. V lhw for h 7. x y b for x z Find
More informationSYSTEMS OF LINEAR EQUATIONS
SYSTEMS OF LINEAR EQUATIONS A system of linear equations is a set of two equations of lines. A solution of a system of linear equations is the set of ordered pairs that makes each equation true. That is
More informationLesson Plan #001. Class: Geometry Date: Wednesday September 9 th, 2015
Lesson Plan #001 1 Class: Geometry Date: Wednesday September 9 th, 2015 Topic: Points, lines, and planes Aim: What are points, lines and planes? Objectives: 1) Students will be able to describe what is
More informationTrig Functions, Equations & Identities May a. [2 marks] Let. For what values of x does Markscheme (M1)
Trig Functions, Equations & Identities May 2008-2014 1a. Let. For what values of x does () 1b. [5 marks] not exist? Simplify the expression. EITHER OR [5 marks] 2a. 1 In the triangle ABC,, AB = BC + 1.
More informationGEOMETRY APPLICATIONS
GEOMETRY APPLICATIONS Chapter 3: Parallel & Perpendicular Lines Name: Teacher: Pd: 0 Table of Contents DAY 1: (Ch. 3-1 & 3-2) SWBAT: Identify parallel, perpendicular, and skew lines. Identify the angles
More informationAnswers to practice questions for Midterm 1
Answers to practice questions for Midterm Paul Hacking /5/9 (a The RREF (reduced row echelon form of the augmented matrix is So the system of linear equations has exactly one solution given by x =, y =,
More informationStudy Guide and Review
Fill in the blank in each sentence with the vocabulary term that best completes the sentence 1 A is a flat surface made up of points that extends infinitely in all directions A plane is a flat surface
More informationSpring 2012 Student Performance Analysis
Spring 2012 Student Performance Analysis Geometry Standards of Learning Presentation may be paused and resumed using the arrow keys or the mouse. 1 Translating a Short Verbal Argument into Symbolic Form
More informationHartmann HONORS Geometry Chapter 3 Formative Assessment * Required
Hartmann HONORS Geometry Chapter 3 Formative Assessment * Required 1. First Name * 2. Last Name * Vocabulary Match the definition to the vocabulary word. 3. Non coplanar lines that do not intersect. *
More informationGraded Assignment 2 Maple plots
Graded Assignment 2 Maple plots The Maple part of the assignment is to plot the graphs corresponding to the following problems. I ll note some syntax here to get you started see tutorials for more. Problem
More informationMATRIX REVIEW PROBLEMS: Our matrix test will be on Friday May 23rd. Here are some problems to help you review.
MATRIX REVIEW PROBLEMS: Our matrix test will be on Friday May 23rd. Here are some problems to help you review. 1. The intersection of two non-parallel planes is a line. Find the equation of the line. Give
More informationMATH Additional Examples Page 4.24
MAH 050 4.4 Additional Examples Page 4.4 4.4 Additional Examples for Chapter 4 Example 4.4. Prove that the line joining the midpoints of two sides of a triangle is parallel to and exactly half as long
More information3.5 Equations of Lines and Planes
3.5 Equations of Lines and Planes Objectives Iknowhowtodefinealineinthree-dimensionalspace. I can write a line as a parametric equation, a symmetric equation, and a vector equation. I can define a plane
More informationThe mathematics behind projections
The mathematics behind projections This is an article from my home page: www.olewitthansen.dk Ole Witt-Hansen 2005 (2015) Contents 1. The mathematics behind projections and 3-dim graphics...1 1.1 Central
More informationViewing with Computers (OpenGL)
We can now return to three-dimension?', graphics from a computer perspective. Because viewing in computer graphics is based on the synthetic-camera model, we should be able to construct any of the classical
More information(1) Page #1 24 all. (2) Page #7-21 odd, all. (3) Page #8 20 Even, Page 35 # (4) Page #1 8 all #13 23 odd
Geometry/Trigonometry Unit 1: Parallel Lines Notes Name: Date: Period: # (1) Page 25-26 #1 24 all (2) Page 33-34 #7-21 odd, 23 28 all (3) Page 33-34 #8 20 Even, Page 35 #40 44 (4) Page 60 61 #1 8 all #13
More informationGeometry Unit 5 Geometric and Algebraic Connections. Table of Contents
Geometry Unit 5 Geometric and Algebraic Connections Table of Contents Lesson 5 1 Lesson 5 2 Distance.p. 2-3 Midpoint p. 3-4 Partitioning a Directed Line. p. 5-6 Slope. p.7-8 Lesson 5 3 Revisit: Graphing
More information1.5 Part - 2 Inverse Relations and Inverse Functions
1.5 Part - 2 Inverse Relations and Inverse Functions What happens when we reverse the coordinates of all the ordered pairs in a relation? We obviously get another relation, but does it have any similarities
More information8. The triangle is rotated around point D to create a new triangle. This looks like a rigid transformation.
2.1 Transformations in the Plane 1. True 2. True 3. False 4. False 5. True 6. False 7. True 8. The triangle is rotated around point D to create a new triangle. This looks like a rigid transformation. 9.
More informationWHAT YOU SHOULD LEARN
GRAPHS OF EQUATIONS WHAT YOU SHOULD LEARN Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs of equations. Find equations of and sketch graphs of
More informationLines That Intersect Circles
LESSON 11-1 Lines That Intersect Circles Lesson Objectives (p. 746): Vocabulary 1. Interior of a circle (p. 746): 2. Exterior of a circle (p. 746): 3. Chord (p. 746): 4. Secant (p. 746): 5. Tangent of
More information1 EquationsofLinesandPlanesin 3-D
1 EquationsofLinesandPlanesin 3-D Recall that given a point P (a, b, c), one can draw a vector from the origin to P. Such a vector is called the position vector of the point P and its coordinates are a,
More informationvolume & surface area of a right circular cone cut by a plane parallel to symmetrical axis (Hyperbolic section)
From the SelectedWorks of Harish Chandra Rajpoot H.C. Rajpoot Winter December 25, 2016 volume & surface area of a right circular cone cut by a plane parallel to symmetrical axis (Hyperbolic section) Harish
More informationLet s write this out as an explicit equation. Suppose that the point P 0 = (x 0, y 0, z 0 ), P = (x, y, z) and n = (A, B, C).
4. Planes and distances How do we represent a plane Π in R 3? In fact the best way to specify a plane is to give a normal vector n to the plane and a point P 0 on the plane. Then if we are given any point
More informationGeometry Midterm Review Vocabulary:
Name Date Period Geometry Midterm Review 2016-2017 Vocabulary: 1. Points that lie on the same line. 1. 2. Having the same size, same shape 2. 3. These are non-adjacent angles formed by intersecting lines.
More informationTrue/False. MATH 1C: SAMPLE EXAM 1 c Jeffrey A. Anderson ANSWER KEY
MATH 1C: SAMPLE EXAM 1 c Jeffrey A. Anderson ANSWER KEY True/False 10 points: points each) For the problems below, circle T if the answer is true and circle F is the answer is false. After you ve chosen
More informationClipping and Intersection
Clipping and Intersection Clipping: Remove points, line segments, polygons outside a region of interest. Need to discard everything that s outside of our window. Point clipping: Remove points outside window.
More informationGeometry Definitions, Postulates, and Theorems. Chapter 3: Parallel and Perpendicular Lines. Section 3.1: Identify Pairs of Lines and Angles.
Geometry Definitions, Postulates, and Theorems Chapter : Parallel and Perpendicular Lines Section.1: Identify Pairs of Lines and Angles Standards: Prepare for 7.0 Students prove and use theorems involving
More informationSegment Addition Postulate: If B is BETWEEN A and C, then AB + BC = AC. If AB + BC = AC, then B is BETWEEN A and C.
Ruler Postulate: The points on a line can be matched one to one with the REAL numbers. The REAL number that corresponds to a point is the COORDINATE of the point. The DISTANCE between points A and B, written
More information7/7/2016 Unit 4: Linear Relations Grade 9 Mathematics
Rene Descartes, a mathematician who lived during the 17 th century, developed a system for graphing ordered pairs on a grid. This system is called the Cartesian Coordinate System. 1 In this system, ordered
More information3 CHAPTER. Coordinate Geometry
3 CHAPTER We are Starting from a Point but want to Make it a Circle of Infinite Radius Cartesian Plane Ordered pair A pair of numbers a and b instead in a specific order with a at the first place and b
More informationUnit 6: Connecting Algebra and Geometry Through Coordinates
Unit 6: Connecting Algebra and Geometry Through Coordinates The focus of this unit is to have students analyze and prove geometric properties by applying algebraic concepts and skills on a coordinate plane.
More informationIntersecting Simple Surfaces. Dr. Scott Schaefer
Intersecting Simple Surfaces Dr. Scott Schaefer 1 Types of Surfaces Infinite Planes Polygons Convex Ray Shooting Winding Number Spheres Cylinders 2/66 Infinite Planes Defined by a unit normal n and a point
More informationQuaternions and Dual Coupled Orthogonal Rotations in Four-Space
Quaternions and Dual Coupled Orthogonal Rotations in Four-Space Kurt Nalty January 8, 204 Abstract Quaternion multiplication causes tensor stretching) and versor turning) operations. Multiplying by unit
More informationBoardworks Ltd KS3 Mathematics. S1 Lines and Angles
1 KS3 Mathematics S1 Lines and Angles 2 Contents S1 Lines and angles S1.1 Labelling lines and angles S1.2 Parallel and perpendicular lines S1.3 Calculating angles S1.4 Angles in polygons 3 Lines In Mathematics,
More informationFive-Minute Check CCSS Then/Now New Vocabulary Key Concept: Undefined Terms Example 1: Name Lines and Planes Example 2: Real-World Example: Model
Five-Minute Check CCSS Then/Now New Vocabulary Key Concept: Undefined Terms Example 1: Name Lines and Planes Example 2: Real-World Example: Model Points, Lines, and Planes Example 3: Draw Geometric Figures
More informationds dt ds 1 dt 1 dt v v v dt ds and the normal vector is given by N
Normal Vectors and Curvature In the last section, we stated that reparameterization by arc length would help us analyze the twisting and turning of a curve. In this section, we ll see precisely how to
More informationFunctions of Several Variables
Functions of Several Variables Directional Derivatives and the Gradient Vector Philippe B Laval KSU April 7, 2012 Philippe B Laval (KSU) Functions of Several Variables April 7, 2012 1 / 19 Introduction
More informationDirectional Derivatives and the Gradient Vector Part 2
Directional Derivatives and the Gradient Vector Part 2 Lecture 25 February 28, 2007 Recall Fact Recall Fact If f is a dierentiable function of x and y, then f has a directional derivative in the direction
More informationEquations of planes in
Roberto s Notes on Linear Algebra Chapter 6: Lines, planes and other straight objects Section Equations of planes in What you need to know already: What vectors and vector operations are. What linear systems
More informationPractice problems. 1. Given a = 3i 2j and b = 2i + j. Write c = i + j in terms of a and b.
Practice problems 1. Given a = 3i 2j and b = 2i + j. Write c = i + j in terms of a and b. 1, 1 = c 1 3, 2 + c 2 2, 1. Solve c 1, c 2. 2. Suppose a is a vector in the plane. If the component of the a in
More informationGeometry Reasons for Proofs Chapter 1
Geometry Reasons for Proofs Chapter 1 Lesson 1.1 Defined Terms: Undefined Terms: Point: Line: Plane: Space: Postulate 1: Postulate : terms that are explained using undefined and/or other defined terms
More informationColumn and row space of a matrix
Column and row space of a matrix Recall that we can consider matrices as concatenation of rows or columns. c c 2 c 3 A = r r 2 r 3 a a 2 a 3 a 2 a 22 a 23 a 3 a 32 a 33 The space spanned by columns of
More informationL13-Mon-3-Oct-2016-Sec-1-1-Dist-Midpt-HW Graph-HW12-Moodle-Q11, page 1 L13-Mon-3-Oct-2016-Sec-1-1-Dist-Midpt-HW Graph-HW12-Moodle-Q11
L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11, page 1 L13-Mon-3-Oct-016-Sec-1-1-Dist-Midpt-HW11-1--Graph-HW1-Moodle-Q11 1.1 Rectangular Coordinate System: Suppose we know the sum of
More informationEuclid s Axioms. 1 There is exactly one line that contains any two points.
11.1 Basic Notions Euclid s Axioms 1 There is exactly one line that contains any two points. Euclid s Axioms 1 There is exactly one line that contains any two points. 2 If two points line in a plane then
More informationFrom the SelectedWorks of Harish Chandra Rajpoot H.C. Rajpoot. Harish Chandra Rajpoot Rajpoot, HCR. Summer April 6, 2015
From the SelectedWorks of Harish Chandra Rajpoot H.C. Rajpoot Summer April 6, 2015 Mathematical Analysis of Uniform Polyhedron Trapezohedron having 2n congruent right kite faces, 4n edges & 2n+2 vertices
More information1. The Pythagorean Theorem
. The Pythagorean Theorem The Pythagorean theorem states that in any right triangle, the sum of the squares of the side lengths is the square of the hypotenuse length. c 2 = a 2 b 2 This theorem can be
More informationUnit 5: Quadratic Functions
Unit 5: Quadratic Functions LESSON #5: THE PARABOLA GEOMETRIC DEFINITION DIRECTRIX FOCUS LATUS RECTUM Geometric Definition of a Parabola Quadratic Functions Geometrically, a parabola is the set of all
More information2.1 Length of a Line Segment
.1 Length of a Line Segment MATHPOWER TM 10 Ontario Edition pp. 66 7 To find the length of a line segment joining ( 1 y 1 ) and ( y ) use the formula l= ( ) + ( y y ). 1 1 Name An equation of the circle
More informationProperties of Quadratic functions
Name Today s Learning Goals: #1 How do we determine the axis of symmetry and vertex of a quadratic function? Properties of Quadratic functions Date 5-1 Properties of a Quadratic Function A quadratic equation
More informationInversive Plane Geometry
Inversive Plane Geometry An inversive plane is a geometry with three undefined notions: points, circles, and an incidence relation between points and circles, satisfying the following three axioms: (I.1)
More informationProperties of a Function s Graph
Section 3.2 Properties of a Function s Graph Objective 1: Determining the Intercepts of a Function An intercept of a function is a point on the graph of a function where the graph either crosses or touches
More informationLet and be a differentiable function. Let Then be the level surface given by
Module 12 : Total differential, Tangent planes and normals Lecture 35 : Tangent plane and normal [Section 35.1] > Objectives In this section you will learn the following : The notion tangent plane to a
More informationADVANCED EXERCISE 09B: EQUATION OF STRAIGHT LINE
ADVANCED EXERCISE 09B: EQUATION OF STRAIGHT LINE It is given that the straight line L passes through A(5, 5) and is perpendicular to the straight line L : x+ y 5= 0 (a) Find the equation of L (b) Find
More informationNotes Formal Geometry Chapter 3 Parallel and Perpendicular Lines
Name Date Period Notes Formal Geometry Chapter 3 Parallel and Perpendicular Lines 3-1 Parallel Lines and Transversals and 3-2 Angles and Parallel Lines A. Definitions: 1. Parallel Lines: Coplanar lines
More informationOutcomes List for Math Multivariable Calculus (9 th edition of text) Spring
Outcomes List for Math 200-200935 Multivariable Calculus (9 th edition of text) Spring 2009-2010 The purpose of the Outcomes List is to give you a concrete summary of the material you should know, and
More information3 John likes to experiment with geometric. 4 Which of the following conjectures is true for
1 Rectangle ABCD is drawn on a coordinate plane. Each angle measures 90. The rectangle is reflected across the y axis, translated 9 units down, and then rotated 180 clockwise about the origin. What would
More informationDD2429 Computational Photography :00-19:00
. Examination: DD2429 Computational Photography 202-0-8 4:00-9:00 Each problem gives max 5 points. In order to pass you need about 0-5 points. You are allowed to use the lecture notes and standard list
More information5. In the Cartesian plane, a line runs through the points (5, 6) and (-2, -2). What is the slope of the line?
Slope review Using two points to find the slope In mathematics, the slope of a line is often called m. We can find the slope if we have two points on the line. We'll call the first point and the second
More informationDirectional Derivatives and the Gradient Vector Part 2
Directional Derivatives and the Gradient Vector Part 2 Marius Ionescu October 26, 2012 Marius Ionescu () Directional Derivatives and the Gradient Vector Part October 2 26, 2012 1 / 12 Recall Fact Marius
More informationLet s Get This Started!
Lesson. Skills Practice Name Date Let s Get This Started! Points, Lines, Planes, Rays, and Line Segments Vocabulary Write the term that best completes each statement.. A geometric figure created without
More informationUNIT 3 EXPRESSIONS AND EQUATIONS Lesson 3: Creating Quadratic Equations in Two or More Variables
Guided Practice Example 1 Find the y-intercept and vertex of the function f(x) = 2x 2 + x + 3. Determine whether the vertex is a minimum or maximum point on the graph. 1. Determine the y-intercept. The
More informationVirginia Mathematics Checkpoint Assessment GEOMETRY G.3. Topic: Reasoning, Lines, and Transformations
Virginia Mathematics Checkpoint Assessment GEOMETRY G.3 Topic: Reasoning, Lines, and Transformations Standards of Learning Blueprint Summary Reporting Category Geometry SOL Number of Items Reasoning, Lines,
More informationMath 8 Honors Coordinate Geometry part 3 Unit Updated July 29, 2016
Review how to find the distance between two points To find the distance between two points, use the Pythagorean theorem. The difference between is one leg and the difference between and is the other leg.
More informationVideos, Constructions, Definitions, Postulates, Theorems, and Properties
Videos, Constructions, Definitions, Postulates, Theorems, and Properties Videos Proof Overview: http://tinyurl.com/riehlproof Modules 9 and 10: http://tinyurl.com/riehlproof2 Module 9 Review: http://tinyurl.com/module9livelesson-recording
More informationAnnouncements Wednesday, August 23
Announcements Wednesday, August 23 Everything you ll need to know is on the master website: http://people.math.gatech.edu/~cjankowski3/teaching/f2017/m1553/index.html or on the website for this section:
More informationDid You Find a Parking Space?
Lesson.4 Skills Practice Name Date Did You Find a Parking Space? Parallel and Perpendicular Lines on the Coordinate Plane Vocabulary Complete the sentence. 1. The point-slope form of the equation of the
More informationHow to Construct a Perpendicular to a Line (Cont.)
Geometric Constructions How to Construct a Perpendicular to a Line (Cont.) Construct a perpendicular line to each side of this triangle. Find the intersection of the three perpendicular lines. This point
More informationPractice problems from old exams for math 233
Practice problems from old exams for math 233 William H. Meeks III October 26, 2012 Disclaimer: Your instructor covers far more materials that we can possibly fit into a four/five questions exams. These
More informationI can identify, name, and draw points, lines, segments, rays, and planes. I can apply basic facts about points, lines, and planes.
Page 1 of 9 Are You Ready Chapter 1 Pretest & skills Attendance Problems Graph each inequality. 1. x > 3 2. 2 < x < 6 3. x > 1 or x < 0 Vocabulary undefined term point line plane collinear coplanar segment
More informationCHAPTER 5 SYSTEMS OF EQUATIONS. x y
page 1 of Section 5.1 CHAPTER 5 SYSTEMS OF EQUATIONS SECTION 5.1 GAUSSIAN ELIMINATION matrix form of a system of equations The system 2x + 3y + 4z 1 5x + y + 7z 2 can be written as Ax where b 2 3 4 A [
More information3 Solution of Homework
Math 3181 Name: Dr. Franz Rothe February 25, 2014 All3181\3181_spr14h3.tex Homework has to be turned in this handout. The homework can be done in groups up to three due March 11/12 3 Solution of Homework
More informationwhile its direction is given by the right hand rule: point fingers of the right hand in a 1 a 2 a 3 b 1 b 2 b 3 A B = det i j k
I.f Tangent Planes and Normal Lines Again we begin by: Recall: (1) Given two vectors A = a 1 i + a 2 j + a 3 k, B = b 1 i + b 2 j + b 3 k then A B is a vector perpendicular to both A and B. Then length
More informationVectors and the Geometry of Space
Vectors and the Geometry of Space In Figure 11.43, consider the line L through the point P(x 1, y 1, z 1 ) and parallel to the vector. The vector v is a direction vector for the line L, and a, b, and c
More informationBasics of Computational Geometry
Basics of Computational Geometry Nadeem Mohsin October 12, 2013 1 Contents This handout covers the basic concepts of computational geometry. Rather than exhaustively covering all the algorithms, it deals
More information