Review Exercise. 1. Determine vector and parametric equations of the plane that contains the
|
|
|
- Cora Cameron
- 5 years ago
- Views:
Transcription
1 Review Exercise 1. Determine vector and parametric equations of the plane that contains the points A11, 2, 12, B12, 1, 12, and C13, 1, In question 1, there are a variety of different answers possible, depending on the points and direction vectors chosen. Determine two Cartesian equations for this plane using two different vector equations, and verify that these two equations are identical. 3. a. Determine the vector, parametric, and symmetric equations of the line passing through points A1 3, 2, 82 and B14, 3, 92. b. Determine the vector and parametric equations of the plane containing the points A1 3, 2, 82, B14, 3, 92, and C1 2, 1, 32. c. Explain why a symmetric equation cannot exist for a plane. 4. Determine the vector, parametric, and symmetric equations of the line passing through the point A17, 1, 22 and perpendicular to the plane with equation 2x 3y z Determine the Cartesian equation of each of the following planes: a. through the point P10, 1, 22, with normal n! 1 1, 3, 32 b. through the points 13, 0, 12 and 10, 1, 12, and perpendicular to the plane with equation x y z 1 0 c. through the points 11, 2, 12 and 12, 1, 42, and parallel to the x-axis 6. Determine the Cartesian equation of the plane that passes through the origin and contains the line r! 13, 7, 12 t12, 2, 32, t R. 7. Find the vector and parametric equations of the plane that is parallel to the yz-plane and contains the point A1 1, 2, Determine the Cartesian equation of the plane that contains the line r! 12, 3, 22 t11, 1, 42, t R, and the point 14, 3, Determine the Cartesian equation of the plane that contains the following lines: L 1 : r! and L 2 : r! 14, 4, 52 t15, 4, 62, t R, 14, 4, 52 s12, 3, 42, s R 10. Determine an equation for the line that is perpendicular to the plane 3x 2y z 1 passing through 12, 3, 32. Give your answer in vector, parametric, and symmetric form. 11. A plane has 3x 2y z 6 0 as its Cartesian equation. Determine the vector and parametric equations of this plane. 480 REVIEW EXERCISE
2 12. Determine an equation for the line that has the same x- and z-intercepts as the plane with equation 2x 5y z 7 0. Give your answer in vector, parametric, and symmetric form. 13. Determine the vector, parametric, and Cartesian forms of the equation of the plane containing the lines L and! 1 : r! 13, 4, 12 s11, 3, 52, s R, L 2 : r 2 17, 1, 02 t12, 6, 102, t R. 14. Sketch each of the following planes: a. p 1 : 2x 3y 6z 12 0 b. p 2 : 2x 3y 12 0 c. p 3 : x 3z 6 0 d. p 4 : y 2z 4 0 e. p 5 : 2x 3y 6z Determine the vector, parametric, and Cartesian equations of each of the following planes: a. passing through the points and and parallel to the line with equation L : r! P11, 2ti! 2, 52 14t 32j! Q13, 1, 22 1t 12k!, t R b. containing the point A11, 1, 22 and perpendicular to the line joining the points B12, 1, 62 and C1 2, 1, 52 c. passing through the points 14, 1, 12 and 15, 2, 42 and parallel to the z-axis d. passing through the points 11, 3, 52, 12, 6, 42, and 13, 3, Show that s, and L 2 : r! L 1 : r! 11, 2, 32 s1 3, 5, 212 t10, 1, 32, t R, 11, 1, 62 u11, 1, 12 v12, 5, 112, u, v R, are equations for the same plane. 17. The two lines and L 2 : r! L 1 : r! 1 1, 1, 02 s12, 1, 12, s R, 12, 1, 22 t12, 1, 12, t R, are parallel but do not coincide. The point A15, is on Determine the coordinates of a point B on such that AB! 4, 32 L 1. L 2 is perpendicular to L Write a brief description of each plane. a. p 1 : 2x 3y 6 b. p 2 : x 3z 6 c. p 3 : 2y z a. Which of the following points lies on the line x 2t, y 3 t, z 1 t? A12, 4, 22 B1 2, 2, 12 C14, 5, 22 D16, 6, 22 b. If the point 1a, b, 32 lies on the line, determine the values of a and b. CHAPTER 8 481
3 20. Calculate the acute angle that is formed by the intersection of each pair of lines. a. L and L 2 : x 2 1 y 1 : x 1 y b. y 4x 2 and y x 3 c. L 1 : x 1 3t, y 1 4t, z 2t and L 2 : x 1 2s, y 3s, z 7 s d. L 1 : 1x, y, z2 14, 7, 12 t14, 8, 42 and L 2 : 1x, y, z2 11, 5, 42 s1 1, 2, Calculate the acute angle that is formed by the intersection of each pair of planes. a. 2x 3y z 9 0 and x 2y 4 0 b. x y z 1 0 and 2x 3y z a. Which of the following lines is parallel to the plane i. r! 4x y z 10 0? 13, 0, 22 t11, 2, 22 ii. x 3t, y 5 2t, z 10t x 1 iii. y z 1 b. Do any of these lines lie in the plane in part a.? 23. Does the point (4, 5, 6) lie in the plane 1x, y, z2 14, 1, 62 p13, 2, 12 q1 6, 6, 12? Support your answer with the appropriate calculations. 24. Determine the parametric equations of the plane that contains the following two parallel lines: L 1 : 1x, y, z2 12, 4, 12 t13, 1, 12 and L 2 : 1x, y, z2 11, 4, 42 t1 6, 2, Explain why the vector equation of a plane has two parameters, while the vector equation of a line has only one. 26. Explain why any plane with a vector equation of the form 1x, y, z2 1a, b, c2 s1d, e, f 2 t1a, b, c2 will always pass through the origin. 27. a. Explain why the three points 12, 3, 12, 18, 5, 52 and 1 1, 2, 12 do not determine a plane. b. Explain why the line r! 14, 9, 32 t11, 4, 22 and the point 18, 7, 52 do not determine a plane. 28. Find a formula for the scalar equation of a plane in terms of a, b, and c, where a, b, and c are the x-intercept, the y-intercept, and the z-intercept of a plane, respectively. Assume that all intercepts are nonzero. 482 REVIEW EXERCISE
4 29. Determine the Cartesian equation of the plane that has normal vector 16, 5, 122 and passes through the point 15, 8, A plane passes through the points A11, 3, 22, B1 2, 4, 22, and C13, 2, 12. a. Determine a vector equation of the plane. b. Determine a set of parametric equations of the plane. c. Determine the Cartesian equation of the plane. d. Determine if the point 13, 5, 42 lies on the plane. 31. Determine the Cartesian equation of the plane that is parallel to the plane 4x 2y 5z 10 0 and passes through each point below. a. 10, 0, 02 b. 1 1, 5, 12 c. 12, 2, Show that the following pairs of lines intersect. Determine the coordinates of the point of intersection and the angles formed by the lines. a. L 1 : x 5 2t and L 2 : x 23 2s y 3 t y 6 s b. L and L 2 : x 6 y 2 1 : x 3 y Determine the vector equation, parametric equations, and, if possible, symmetric equation of the line that passes through the point P11, 3, 52 and a. has direction vector 1 2, 4, 102 b. also passes through the point Q1 7, 9, 32 c. is parallel to the line that passes through R14, 8, 52 and S1 2, 5, 92 d. is parallel to the x-axis e. is perpendicular to the line 1x, y, z2 11, 0, 52 t1 3, 4, 62 f. is perpendicular to the plane determined by the points A14, 2, 12, B13, 4, 22, and C1 3, 2, Determine the Cartesian equation of the plane that a. contains the point P1 2, 6, 12 and has normal vector 12, 4, 52 x 4 b. contains the point and the line y 2 z 1 P1 2, 0, c. contains the point P13, 3, 32 and is parallel to the xy-plane d. contains the point P1 4, 2, 42 and is parallel to the plane 3x y 4z 8 0 e. is perpendicular to the yz-plane and has y-intercept 4 and z-intercept 2 f. is perpendicular to the plane x 2y z 6 and contains the line 1x, y, z2 12, 1, 12 t13, 1, 22 CHAPTER 8 483
5 Chapter 8 Test 1. a. Given the points A11, 2, 42, B12, 0, 32, and C14, 4, 42, i. determine the vector and parametric equations of the plane that contains these three points ii. determine the corresponding Cartesian equation of the plane that contains these three points b. Does the point with coordinates Q1, 1, 1 lie on this plane? 2 R 2. The plane p intersects the coordinate axes at 12, 0, 02, 10, 3, 02, and 10, 0, 42. x a. Write an equation for this plane, expressing it in the form a y b z 1. c b. Determine the coordinates of a normal to this plane. 3. a. Determine a vector equation for the plane containing the origin and the line with equation r! 12, 1, 32 t11, 2, 52, t R. b. Determine the corresponding Cartesian equation of this plane. 4. a. Determine a vector equation for the plane that contains the following two lines: L 1 : r! and L 2 : r! 14, 3, 52 t12, 0, 32, t R, 14, 3, 52 s15, 1, 12, s R b. Determine the corresponding Cartesian equation of this plane. x 2 5. a. A line has y 4 as its symmetric equations. Determine the 4 2 z coordinates of the point where this line intersects the yz-plane. b. Write a second symmetric equation for this line using the point you found in part a. 6. a. Determine the angle between p 1 and p 2 where the two planes are defined as p 1 : x y z 0 and p 2 : x y z 0. b. Given the planes p 3 : 2x y kz 5 and p 4 : kx 2y 8z 9, i. determine a value of k if these planes are parallel ii. determine a value of k if these planes are perpendicular c. Explain why the two given equations that contain the parameter k in part b cannot represent two identical planes. 7. a. Using a set of coordinate axes in R 2, sketch the line x 2y 0. b. Using a set of coordinate axes in R 3, sketch the plane x 2y 0. c. The equation Ax By 0, A, B 0, represents an equation of a plane in R 3. Explain why this plane must always contain the z-axis. 484 CHAPTER 8 TEST
6
7
Section 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
Section 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,
WHAT 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
Suggested problems - solutions
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
The 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
Each point P in the xy-plane corresponds to an ordered pair (x, y) of real numbers called the coordinates of P.
Lecture 7, Part I: Section 1.1 Rectangular Coordinates Rectangular or Cartesian coordinate system Pythagorean theorem Distance formula Midpoint formula Lecture 7, Part II: Section 1.2 Graph of Equations
.(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,
7.3 3-D Notes Honors Precalculus Date: Adapted from 11.1 & 11.4
73 3-D Notes Honors Precalculus Date: Adapted from 111 & 114 The Three-Variable Coordinate System I Cartesian Plane The familiar xy-coordinate system is used to represent pairs of numbers (ordered pairs
Vectors. 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
Mathematics (www.tiwariacademy.com)
() Miscellaneous Exercise on Chapter 10 Question 1: Find the values of k for which the line is (a) Parallel to the x-axis, (b) Parallel to the y-axis, (c) Passing through the origin. Answer 1: The given
Chapter 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
Graphs of Equations. MATH 160, Precalculus. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Graphs of Equations
Graphs of Equations MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: sketch the graphs of equations, find the x- and y-intercepts
Geometry 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
Section 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
Section 2.1 Graphs. The Coordinate Plane
Section 2.1 Graphs The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of numbers to form
1.8 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
CHAPTER - 10 STRAIGHT LINES Slope or gradient of a line is defined as m = tan, ( 90 ), where is angle which the line makes with positive direction of x-axis measured in anticlockwise direction, 0 < 180
1.5 Equations of Lines and Planes in 3-D
1.5. EQUATIONS OF LINES AND PLANES IN 3-D 55 Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3-D Recall that given a point P = (a, b, c), one can draw a vector from the
JUST THE MATHS SLIDES NUMBER 5.2. GEOMETRY 2 (The straight line) A.J.Hobson
JUST THE MATHS SLIDES NUMBER 5.2 GEOMETRY 2 (The straight line) by A.J.Hobson 5.2.1 Preamble 5.2.2 Standard equations of a straight line 5.2.3 Perpendicular straight lines 5.2.4 Change of origin UNIT 5.2
Problems 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
THREE DIMENSIONAL GEOMETRY
For more important questions visit : www4onocom CHAPTER 11 THREE DIMENSIONAL GEOMETRY POINTS TO REMEMBER Distance between points P(x 1 ) and Q(x, y, z ) is PQ x x y y z z 1 1 1 (i) The coordinates of point
Updated: 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,
Math (Spring 2009): Lecture 5 Planes. Parametric equations of curves and lines
Math 18.02 (Spring 2009): Lecture 5 Planes. Parametric equations of curves and lines February 12 Reading Material: From Simmons: 17.1 and 17.2. Last time: Square Systems. Word problem. How many solutions?
Vectors 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
Lines 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
Rectangular Coordinates in Space
Rectangular Coordinates in Space Philippe B. Laval KSU Today Philippe B. Laval (KSU) Rectangular Coordinates in Space Today 1 / 11 Introduction We quickly review one and two-dimensional spaces and then
5. 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
Properties 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
f( x ), or a solution to the equation f( x) 0. You are already familiar with ways of solving
The Bisection Method and Newton s Method. If f( x ) a function, then a number r for which f( r) 0 is called a zero or a root of the function f( x ), or a solution to the equation f( x) 0. You are already
The points (2, 2, 1) and (0, 1, 2) are graphed below in 3-space:
Three-Dimensional Coordinate Systems The plane is a two-dimensional coordinate system in the sense that any point in the plane can be uniquely described using two coordinates (usually x and y, but we have
True/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
CHAPTER 2 REVIEW COORDINATE GEOMETRY MATH Warm-Up: See Solved Homework questions. 2.2 Cartesian coordinate system
CHAPTER 2 REVIEW COORDINATE GEOMETRY MATH6 2.1 Warm-Up: See Solved Homework questions 2.2 Cartesian coordinate system Coordinate axes: Two perpendicular lines that intersect at the origin O on each line.
UNIT NUMBER 5.2. GEOMETRY 2 (The straight line) A.J.Hobson
JUST THE MATHS UNIT NUMBER 5.2 GEOMETRY 2 (The straight line) b A.J.Hobson 5.2.1 Preamble 5.2.2 Standard equations of a straight line 5.2. Perpendicular straight lines 5.2.4 Change of origin 5.2.5 Exercises
Coordinate Geometry. Coordinate geometry is the study of the relationships between points on the Cartesian plane
Coordinate Geometry Coordinate geometry is the study of the relationships between points on the Cartesian plane What we will explore in this tutorial (a) Explore gradient I. Identify the gradient of a
Answers 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 =,
12.5 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
GEOMETRY 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...............................................
Pre-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:
1.5 Equations of Lines and Planes in 3-D
56 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 3-D Recall that given a point P = (a, b, c), one can draw a vector from
Multivariable Calculus
Multivariable Calculus Chapter 10 Topics in Analytic Geometry (Optional) 1. Inclination of a line p. 5. Circles p. 4 9. Determining Conic Type p. 13. Angle between lines p. 6. Parabolas p. 5 10. Rotation
Def.: a, b, and c are called the for the line L. x = y = z =
Bob Brown, CCBC Dundalk Math 253 Calculus 3, Chapter Section 5 Completed Lines in Space Eercise : Consider the vector v = Sketch and describe the following set: t v ta, tb, tc : t a, b, c. Let P =,,. Sketch
Chapter P: Preparation for Calculus
1. Which of the following is the correct graph of y = x x 3? E) Copyright Houghton Mifflin Company. All rights reserved. 1 . Which of the following is the correct graph of y = 3x x? E) Copyright Houghton
3, 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
Three Dimensional Geometry. Linear Programming
Three Dimensional Geometry Linear Programming A plane is determined uniquely if any one of the following is known: The normal to the plane and its distance from the origin is given, i.e. equation of a
Writing 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
UNIT NUMBER 5.6. GEOMETRY 6 (Conic sections - the parabola) A.J.Hobson
JUST THE MATHS UNIT NUMBER 5.6 GEMETRY 6 (Conic sections - the parabola) b A.J.Hobson 5.6.1 Introduction (the standard parabola) 5.6.2 ther forms of the equation of a parabola 5.6. Exercises 5.6.4 Answers
with slopes m 1 and m 2 ), if and only if its coordinates satisfy the equation y y 0 = 0 and Ax + By + C 2
CHAPTER 10 Straight lines Learning Objectives (i) Slope (m) of a non-vertical line passing through the points (x 1 ) is given by (ii) If a line makes an angle α with the positive direction of x-axis, then
Functions. Copyright Cengage Learning. All rights reserved.
Functions Copyright Cengage Learning. All rights reserved. 2.2 Graphs Of Functions Copyright Cengage Learning. All rights reserved. Objectives Graphing Functions by Plotting Points Graphing Functions with
L13-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
3-DIMENSIONAL GEOMETRY. 1. Find the distance between the planes, 2x + 3y +4z =4 and 4x +6y +8z =12
3-DIMENSIONAL GEOMETRY 1 Find the distance between the planes, 2x + 3y +4z 4 and 4x +6y +8z 12 Ans 2 29 2Find the length of the perpendicular drawn from the origin to the plane 2x- 3y+6z+21 0 Ans 3 7 unit
Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations
Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations origin (x, y) Ordered pair (x-coordinate, y-coordinate) (abscissa, ordinate) x axis Rectangular or
You should be able to plot points on the coordinate axis. You should know that the the midpoint of the line segment joining (x, y 1 1
Name GRAPHICAL REPRESENTATION OF DATA: You should be able to plot points on the coordinate axis. You should know that the the midpoint of the line segment joining (x, y 1 1 ) and (x, y ) is x1 x y1 y,.
Section 3.1 Objective 1: Plot Points in the Rectangular Coordinate System Video Length 12:35
Section 3.1 Video Guide The Rectangular Coordinate System and Equations in Two Variables Objectives: 1. Plot Points in the Rectangular Coordinate System 2. Determine If an Ordered Pair Satisfies an Equation
Unit 12 Topics in Analytic Geometry - Classwork
Unit 1 Topics in Analytic Geometry - Classwork Back in Unit 7, we delved into the algebra and geometry of lines. We showed that lines can be written in several forms: a) the general form: Ax + By + C =
UNIT 4 NOTES. 4-1 and 4-2 Coordinate Plane
UNIT 4 NOTES 4-1 and 4-2 Coordinate Plane y Ordered pairs on a graph have several names. (X coordinate, Y coordinate) (Domain, Range) (Input,Output) Plot these points and label them: a. (3,-4) b. (-5,2)
TECHNISCHE UNIVERSITÄT BERLIN WS2005 Fachbereich 3 - Mathematik Prof. Dr. U. Pinkall / Charles Gunn Abgabe:
TECHNISCHE UNIVERSITÄT BERLIN WS2005 Fachbereich 3 - Mathematik Prof. Dr. U. Pinkall / Charles Gunn Abgabe: 27.10.2005 1. Übung Geometrie I (Lines in P 2 (R), Separation, Desargues) 1. Aufgabe Line equations
slope rise run Definition of Slope
The Slope of a Line Mathematicians have developed a useful measure of the steepness of a line, called the slope of the line. Slope compares the vertical change (the rise) to the horizontal change (the
1 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,
Section 2.2 Graphs of Linear Functions
Section. Graphs of Linear Functions Section. Graphs of Linear Functions When we are working with a new function, it is useful to know as much as we can about the function: its graph, where the function
Lesson 5.6: Angles in Standard Position
Lesson 5.6: Angles in Standard Position IM3 - Santowski IM3 - Santowski 1 Fast Five Opening Exercises! Use your TI 84 calculator:! Evaluate sin(50 ) " illustrate with a diagram! Evaluate sin(130 ) " Q
Polar Coordinates. Chapter 10: Parametric Equations and Polar coordinates, Section 10.3: Polar coordinates 27 / 45
: Given any point P = (x, y) on the plane r stands for the distance from the origin (0, 0). θ stands for the angle from positive x-axis to OP. Polar coordinate: (r, θ) Chapter 10: Parametric Equations
The x coordinate tells you how far left or right from center the point is. The y coordinate tells you how far up or down from center the point is.
We will review the Cartesian plane and some familiar formulas. College algebra Graphs 1: The Rectangular Coordinate System, Graphs of Equations, Distance and Midpoint Formulas, Equations of Circles Section
UNIT 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
Three-Dimensional Coordinate Systems
Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 17 Notes These notes correspond to Section 10.1 in the text. Three-Dimensional Coordinate Systems Over the course of the next several lectures, we will
Revision Topic 11: Straight Line Graphs
Revision Topic : Straight Line Graphs The simplest way to draw a straight line graph is to produce a table of values. Example: Draw the lines y = x and y = 6 x. Table of values for y = x x y - - - - =
GEOMETRY 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
Homework Questions 1 Gradient of a Line using y=mx+c
(C1-5.1a) Name: Homework Questions 1 Gradient of a Line using y=mx+c 1. State the gradient and the y-intercept of the following linear equations a) y = 2x 3 b) y = 4 6x m= 2 c = -3 c) 2y = 8x + 4 m= -6
CURVE SKETCHING EXAM QUESTIONS
CURVE SKETCHING EXAM QUESTIONS Question 1 (**) a) Express f ( x ) in the form ( ) 2 f x = x + 6x + 10, x R. f ( x) = ( x + a) 2 + b, where a and b are integers. b) Describe geometrically the transformations
Practice 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
ADVANCED 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
2. From General Form: y = ax 2 + bx + c # of x-intercepts determined by the, D =
Alg2H 5-3 Using the Discriminant, x-intercepts, and the Quadratic Formula WK#6 Lesson / Homework --Complete without calculator Read p.181-p.186. Textbook required for reference as well as to check some
Let 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
The Rectangular Coordinate System and Equations of Lines. College Algebra
The Rectangular Coordinate System and Equations of Lines College Algebra Cartesian Coordinate System A grid system based on a two-dimensional plane with perpendicular axes: horizontal axis is the x-axis
SECTION 1.2 (e-book 2.3) Functions: Graphs & Properties
SECTION 1.2 (e-book 2.3) Functions: Graphs & Properties Definition (Graph Form): A function f can be defined by a graph in the xy-plane. In this case the output can be obtained by drawing vertical line
Algebra. Chapter 4: FUNCTIONS. Name: Teacher: Pd:
Algebra Chapter 4: FUNCTIONS Name: Teacher: Pd: Table of Contents Day1: Chapter 4-1: Relations SWBAT: (1) Identify the domain and range of relations and functions (2) Match simple graphs with situations
Rational Numbers: Graphing: The Coordinate Plane
Rational Numbers: Graphing: The Coordinate Plane A special kind of plane used in mathematics is the coordinate plane, sometimes called the Cartesian plane after its inventor, René Descartes. It is one
If the center of the sphere is the origin the the equation is. x y z 2ux 2vy 2wz d 0 -(2)
Sphere Definition: A sphere is the locus of a point which remains at a constant distance from a fixed point. The fixed point is called the centre and the constant distance is the radius of the sphere.
Topic 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
Workbook. MAT 397: Calculus III
Workbook MAT 397: Calculus III Instructor: Caleb McWhorter Name: Summer 2017 Contents Preface..................................................... 2 1 Spatial Geometry & Vectors 3 1.1 Basic n Euclidean
Math 1313 Prerequisites/Test 1 Review
Math 1313 Prerequisites/Test 1 Review Test 1 (Prerequisite Test) is the only exam that can be done from ANYWHERE online. Two attempts. See Online Assignments in your CASA account. Note the deadline too.
Quadric Surfaces. Philippe B. Laval. Today KSU. Philippe B. Laval (KSU) Quadric Surfaces Today 1 / 24
Quadric Surfaces Philippe B. Laval KSU Today Philippe B. Laval (KSU) Quadric Surfaces Today 1 / 24 Introduction A quadric surface is the graph of a second degree equation in three variables. The general
Quadric Surfaces. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) Quadric Surfaces Spring /
.... Quadric Surfaces Philippe B. Laval KSU Spring 2012 Philippe B. Laval (KSU) Quadric Surfaces Spring 2012 1 / 15 Introduction A quadric surface is the graph of a second degree equation in three variables.
12.7 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
Directional Derivatives. Directional Derivatives. Directional Derivatives. Directional Derivatives. Directional Derivatives. Directional Derivatives
Recall that if z = f(x, y), then the partial derivatives f x and f y are defined as and represent the rates of change of z in the x- and y-directions, that is, in the directions of the unit vectors i and
12 - THREE DIMENSIONAL GEOMETRY Page 1 ( Answers at the end of all questions ) = 2. ( d ) - 3. ^i - 2. ^j c 3. ( d )
- THREE DIMENSIONAL GEOMETRY Page ( ) If the angle θ between the line x - y + x + y - z - and the plane λ x + 4 0 is such that sin θ, then the value of λ is - 4-4 [ AIEEE 00 ] ( ) If the plane ax - ay
Drill Exercise - 1. Drill Exercise - 2. Drill Exercise - 3
Drill Exercise -. Find the distance between the pair of points, (a sin, b cos ) and ( a cos, b sin ).. Prove that the points (a, 4a) (a, 6a) and (a + 3 a, 5a) are the vertices of an equilateral triangle.
Geometry Definitions and Theorems. Chapter 9. Definitions and Important Terms & Facts
Geometry Definitions and Theorems Chapter 9 Definitions and Important Terms & Facts A circle is the set of points in a plane at a given distance from a given point in that plane. The given point is the
Lecture 5. If, as shown in figure, we form a right triangle With P1 and P2 as vertices, then length of the horizontal
Distance; Circles; Equations of the form Lecture 5 y = ax + bx + c In this lecture we shall derive a formula for the distance between two points in a coordinate plane, and we shall use that formula to
Drill Exercise - 1. Drill Exercise - 2. Drill Exercise - 3
Drill Exercise - 1 1. Find the distance between the pair of points, (a sin, b cos ) and ( a cos, b sin ). 2. Prove that the points (2a, 4a) (2a, 6a) and (2a + 3 a, 5a) are the vertices of an equilateral
Algebra 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?
GEOMETRY HONORS COORDINATE GEOMETRY PACKET
GEOMETRY HONORS COORDINATE GEOMETRY PACKET Name Period Homework Lesson Assignment Day 1 - Slopes of Perpendicular WKSHT and Parallel Lines Day 2 - Writing an Equation of a Line HW- Honors TXTBK pages 615-617
Math 113 Calculus III Final Exam Practice Problems Spring 2003
Math 113 Calculus III Final Exam Practice Problems Spring 23 1. Let g(x, y, z) = 2x 2 + y 2 + 4z 2. (a) Describe the shapes of the level surfaces of g. (b) In three different graphs, sketch the three cross
PARAMETRIC EQUATIONS AND POLAR COORDINATES
10 PARAMETRIC EQUATIONS AND POLAR COORDINATES PARAMETRIC EQUATIONS & POLAR COORDINATES A coordinate system represents a point in the plane by an ordered pair of numbers called coordinates. PARAMETRIC EQUATIONS
Cylinders and Quadric Surfaces A cylinder is a three dimensional shape that is determined by
Cylinders and Quadric Surfaces A cylinder is a three dimensional shape that is determined by a two dimensional (plane) curve C in three dimensional space a line L in a plane not parallel to the one in
Downloaded from Class XI Chapter 12 Introduction to Three Dimensional Geometry Maths
A point is on the axis. What are its coordinates and coordinates? If a point is on the axis, then its coordinates and coordinates are zero. A point is in the XZplane. What can you say about its coordinate?
INTRODUCTION TO THREE DIMENSIONAL GEOMETRY
Chapter 1 INTRODUCTION TO THREE DIMENSIONAL GEOMETRY Mathematics is both the queen and the hand-maiden of all sciences E.T. BELL 1.1 Introduction You may recall that to locate the position of a point in
6.7. POLAR COORDINATES
6.7. POLAR COORDINATES What You Should Learn Plot points on the polar coordinate system. Convert points from rectangular to polar form and vice versa. Convert equations from rectangular to polar form and
Key Idea. It is not helpful to plot points to sketch a surface. Mainly we use traces and intercepts to sketch
Section 12.7 Quadric surfaces 12.7 1 Learning outcomes After completing this section, you will inshaallah be able to 1. know what are quadric surfaces 2. how to sketch quadric surfaces 3. how to identify
I 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
The Coordinate System and Graphs
The Coordinate System and Graphs - 3.1-3.2 Fall 2013 - Math 1010 (Math 1010) M 1010 3.1-3.2 1 / 17 Roadmap Plotting ordered pairs. The distance formula. The midpoint formula. Graphs of equations. Intercepts.