MAT 1339-S14 Class 10 & 11
|
|
- Dustin Townsend
- 5 years ago
- Views:
Transcription
1 MAT 1339-S14 Class 10 & 11 August 7 & 11, 2014 Contents 8 Lines and Planes Equations of Lines in Two-Space and Three-Space Equations of Planes Properties of Planes Intersection of Lines in Two-Space and Three-Space Intersection of Lines and Planes Intersection of Planes Lines and Planes 8.1 Equations of Lines in Two-Space and Three-Space Lines and planes are geometrical objects. Equations of lines and planes are their algebraic representations. The key to link the geometrical objects and their algebraic equations is that any point on the lines or planes satisfies the corresponding equations. Slope-Intercept Equation In two-sapce, a line can be defined by an equation in slope-intercept form as following y = mx + b where m is the slope of the line and b is the y-intercept. Scalar Equation The scalar equation (also called the standard equation) of a line in two-space is is a normal vector to the line. ax + by + c = 0, where n = [a, b] 1
2 Definition 8.1. A normal vector to a line is a vector that is perpendicular to the line. Example 8.2. Consider the line x + 2y + 1 = 0. How can we check that n = [1, 2] is perpendicular to the line? Solution: Choose any two points on the line, for example (0, 1 ) and ( 1, 0) are two points on 2 the line. Given two points, we get a vector [ m = [x 2 x 1, y 2 y 1 ] = 1, 1 ] 2 and this vector is parallel to the line. The dot product n m = [1, 2] [ 1, 1 ] = 0 implies that n is perpendicular to m. 2 We conclude that the n is perpendicular to the line x + 2y + 1 = 0. Vector Equation The vector equation of a line in two-space is where t R is a scalar, r = r 0 + t m or [x, y] = [x 0, y 0 ] + t[m 1, m 2 ] r = [x, y] is a vector corresponding to any unknown point on the line, r 0 = [x 0, y 0 ] is a vector corresponding to any known point on the line, m = [m 1, m 2 ] is a direction vector parallel to the line. Example 8.3. Write the vector equation of the line x + 2y + 1 = 0. Solution: The direction vector of this line is m = [x 2 x 1, y 2 y 1 ] = [ 1, 1 ] 2 and we know the point ( 1, 0) is on the line. So the vector equation of the line is r = [ 1, 0] + t[ 1, 1 ]. 2 Parametric Equation The parametric equation of a line in two-space is x = x 0 + tm 1 where t R is the parameter. y = y 0 + tm 2 2
3 Example 8.4. Consider the line a) Find two points on the line. b) Write the vector equation of the line. c) Write the scalar equation of the line. Solution: x = 3 + 2t y = 5 + 4t. a) Any values of t will produce a point on the line. Let t = 0, we get a point (3, 5). Let t = 1, we get another point (5, 1). b) The vector equation of the line is [x, y] = [3, 5] + t[2, 4]. c) To get the scalar equation of the line, we need to isolate t in the parametric equation. x = 3 + 2t = t = x 3 2 y = 5 + 4t = t = y x 3 = y + 5 4x 12 = 2y Thus the scalar equation of the line is Equations of Lines in Three-Space 2x y 11 = 0. The vector equation and the parametric equation of lines in two-space can be easily generalized to the vector equation and the parametric equation of lines in three-space. Vector Equation The vector equation of a line in three-space is where r = r 0 + t m or [x, y, z] = [x 0, y 0, z 0 ] + t[m 1, m 2, m 3 ] t R is a scalar, r = [x, y, z] is a vector corresponding to any unknown point on the line, r 0 = [x 0, y 0, z 0 ] is a vector corresponding to any known point on the line, m = [m 1, m 2, m 3 ] is a direction vector parallel to the line. 3
4 Parametric Equation The parametric equation of a line in two-space is x = x 0 + tm 1 where t R is the parameter. y = y 0 + tm 2 z = z 0 + tm 3 Example 8.5. A line passes through points (2, 1, 0) and (1, 0, 3). a) Write the vector equation of the line. b) Write the parametric equation of the line. c) Determine whether the point (3, 1, 2) lies on the line. Solution: a) Given two points (2, 1, 0) and (1, 0, 3), we get the direction vector Thus the vector equation of the line is b) The parametric equation of the line is [1, 0, 3] [2, 1, 0] = [ 1, 1, 3]. [x, y, z] = [2, 1, 0] + t[ 1, 1, 3]. x = 2 t y = 1 t z = 0 + 3t. c) If (3, 1, 2) lies on the line, then there exists some t R such that So [3, 1, 2] = [2, 1, 0] + t[ 1, 1, 3]. 3 = 2 t = t = 1 1 = 1 t = t = 0 2 = 0 + 3t = t = 2 3. The t-values are not equal, thus the point (3, 1, 2) does not lie on the line. Example 8.6. Find the vector equation of the line in two-space which is perpendicular to 4x 3y = 17 and passes through the point ( 2, 4). 4
5 Solution: We know that n = [4, 3] is a normal vector of 4x 3y = 17. So n is perpendicular to 4x 3y = 17 and n gives a direction vector of the line we want. The vector equation of the line is where t R is a scalar. [x, y] = [ 2, 4] + t[4, 3], Example 8.7. Find the parametric equation of the line in three-space which is parallel to the z-axis and passes through the point (1, 5, 10). Solution: The vector [0, 0, 1] is parallel to the z-axis, thus gives a direction vector of the line we want. The parametric equation of the line is Note x = 1 + 0t y = 5 + 0t z = t where t R is a parameter. a) A line in two-space or three-space is determined by two distinct points on the line, or a point on the line and a direction vector parallel to the line. b) There is no slope-intercept equation of a line in three-space because a line in threespace does not necessarily intersect with the axes. c) There is no scalar equation of a line in three-space because a line in three-sapce has infinite number of normal vectors which are not necessarily parallel to one another. 8.2 Equations of Planes Scalar Equation The scalar equation (also called the standard equation) of a plane in three-space is ax + by + cz + d = 0, where n = [a, b, c] is a normal vector to the plane. Definition 8.8. A normal vector to a plane is a vector that is perpendicular to the plane. Example 8.9. Consider the plane x + 2y + z + 1 = 0. How can we check that n = [1, 2, 1] 5
6 is perpendicular to the plane? Choose any two points on the plane, for example (0, 1, 0) and ( 1, 1, 2) are two 2 points on the plane. Given two points, we get a vector m = [x 2 x 1, y 2 y 1, z 2 z 1 ] = [ 1, 32 ], 2 and this vector is parallel to the plane. The dot product n m = [1, 2, 1] [ 1, 32 ], 2 = 0 implies that n is perpendicular to m. So n is perpendicular to the plane x+2y+z+1 = 0. Vector Equation The vector equation of a plane in three-space is r = r 0 + t a + s b or [x, y, z] = [x 0, y 0, z 0 ] + t[a 1, a 2, a 3 ] + s[b 1, b 2, b 3 ] where t, s R are scalars, r = [x, y, z] is a vector corresponding to any unknown point on the line, r 0 = [x 0, y 0, z 0 ] is a vector corresponding to any known point on the line, a = [a 1, a 2, a 3 ] and b = [b 1, b 2, b 3 ] are two non-collienar direction vectors parallel to the plane. Parametric Equation The parametric equation of a plane in three-space is x = x 0 + ta 1 + sb 1 where t, s R are the parameters. y = y 0 + ta 2 + sb 2 z = z 0 + ta 3 + sb 3 Example Consider the plane with direction vectors a = [1, 2, 3] and b = [2, 1, 4] passing through the point (2, 0, 1). a) Write the vector equation of the plane. b) Write the parametric equation of the plane. c) Find two other points on the plane. d) Find the y-intercept of the plane. 6
7 Solution: a) The vector equation of the plane is [x, y, z] = [2, 0, 1] + t[1, 2, 3] + s[2, 1, 4]. b) The parametric equation of the plane is x = 2 + t + 2s y = 0 + 2t s z = 1 3t + 4s. c) Any combination of values of the parameters t and s will produce a point on the plane. Let t = 0 and s = 1, we get a point (4, 1, 5). Let t = 1 and s = 0, we get another point (3, 2, 2). d) In order to find the y-intercept of the plane, we let x = 0 and z = 0 and solve for t and s. Let 0 = 2 + t + 2s 0 = 1 3t + 4s. we get t = 3 5 and s = Then y = 0 + 2t s = Properties of Planes For the previous example Example How to write the scaler equation of the plane! Note: A plane in three-space is determined by three non-collinear points on the plane or by a point on the plane and two non-collienar direction vectors parallel to the plane. 7
8 Example 8.11 (Example 2 page 457). Find the scalar equation of the plane containing the points A( 3, 1, 2), B(4, 6, 2), and C(5, 4, 1). Key concepts: The scalar equation of a plane is three space is Ax + By + Cz + D = 0, where n = [A, B, C] is a normal vector to the plane. Any vector parallel to the normal vector of a plane is also normal to the plane. The coordinates of any point on the plane satisfy the scalar equation. A normal vector (for orientation) and a point (for position) can be used to define a plane. 8
9 Class 11 August 11, Intersection of Lines in Two-Space and Three-Space Intersection of Lines in Two Lines in Two-Space Geometrically there are three possibilities for the intersection of two lines in two-space. Intersect at a point Parallel Coincident Algebraically there are three possibilities for the solutions of the following system of equations. Unique solution No solution Infinitely many solutions a 1 x + b 1 y + c 1 = 0 a 2 x + b 2 y + c 2 = 0 Example (1) The system of equations x y + 1 = 0 2x + y + 4 = 0 9
10 has unique solution x = 5 3 and y = 2 3. We can also write the solution as a point (x, y) = ( 5, 2 ) or as a vector 3 3 [ [x, y] = 5 ] 3, 2. 3 (2) The system of equations x y + 1 = 0 x y + 1 = 0 has infinitely many solutions since all values of x = t, let y = t + 1, then is a solution for the system. (3) The system of equations [t, t + 1] x y + 1 = 0 x y 2 = 0 has no solution. There is no value of x and y satisfying both equations. Intersection of Two Lines in Three-Space Geometrically there are four possibilities for the intersection of two lines in three-space. Intersect at a point Coincident Parallel Skew (not parallel and not intersect) 10
11 Algebraically there are three possibilities for the solutions of the following system of equations. [x, y, z] = [x 0, y 0, z 0 ] + s[m 1, m 2, m 3 ] Unique solution (intersect at a point) Infinitely many solutions (coincident) No solution (parallel or skew) Example The system of equations [x, y, z] = [a 0, b 0, c 0 ] + t[n 1, n 2, n 3 ] [x, y, z] = [7, 2, 6] + s[2, 1, 3] 1 [x, y, z] = [3, 9, 13] + t[1, 5, 5] 2 has a unique solution. From 1 we get [x, y, z] = [7 + 2s, 2 + s, 6 3s]. From 2 we get [x, y, z] = [3 + t, 9 + 5t, t]. Thus we get another system of equations in terms of s and t as following s = 3 + t 2 + s = 9 + 5t 6 3s = t Solve this system: We get s = 3 and t = 2. Thus the unique solution of the original system is [x, y, z] = [7, 2, 6] + ( 3)[2, 1, 3] = [1, 1, 3] and the two lines intersect at the point (1, 1, 3). 11
12 8.5 Intersection of Lines and Planes Geometrically there are three possibilities for the intersection of a line and a plane in three-space. Intersect at a point The line lies on the plane Parallel Algebraically there are three possibilities for the solutions of the following system of equations. [x, y, z] = [x 0, y 0, z 0 ] + t[m 1, m 2, m 3 ] Unique solution (intersect at a point) ax + by + cz + d = 0 Infinitely many solutions (the line lies on the plane) No solution (parallel) Example The system of equations [x, y, z] = [5, 5, 2] + t[2, 5, 3] 9x + 13y 2z =
13 has a unique solution. From 1 we can write the parametric equation of the line x = 5 + 2t y = 5 5t z = 2 + 3t. Substitute the parametric equation into 2 we get 9(5 + 2t) + 13( 5 5t) 2(2 + 3t) = 29. Expand and solve for t we get t = 1. Thus [x, y, z] = [5, 5, 2] + ( 1)[2, 5, 3] = [3, 0, 1] is the unique solution. The line and the plane intersect at the point (3, 0, 1). Example 8.15 (The Distance From a Point to a Plane, Example 3, Page 477). Consider the plane with scalar equation 4x + 2y + z 16 = 0. (a) Determine if the point P = (10, 3, 8) is on the plane. (b) Write the vector and parametric equations of the line l, which is perpendicular to the plane and passes through the point Q = (10, 3, 8). (c) Find the intersection of the line l and the plane. 13
14 (d) Determine the distance between the point Q = (10, 3, 8) and the plane. Exercise: Do the same for the point B = (2, 2, 4) 8.6 Intersection of Planes Intersection of Two Planes in Three-Space Geometrically there are three possibilities for the intersection of two planes in three-space. Intersect in a line Coincident Parallel 14
15 Algebraically there are two possibilities for the solutions of the following system of equations. a 1 x + b 1 y + c 1 z + d 1 = 0 a 2 x + b 2 y + c 2 z + d 2 = 0 Infinitely many solutions (intersect in a line or coincident) No solution (parallel) Example 8.16 (Example 1, Page 483). Describe how the planes in each pair intersect.. a) π 1 : 2x y + z 1 = 0 1 π 2 : x + y + z 6 = 0 2 The normal vector for the plane 1 is n 1 = [2, 1, 1] and the normal vector for the plane 2 is n 2 = [1, 1, 1]. Since n 2 is not a scalar multiple of n 1, we know that n 2 is not parallel to n 1. Thus the two planes must intersect. To solve for the intersection points. We expect the result to be a line of intersection. We want to write the parametric equation and the vector equation of the expected intersection line. Now, 1 2, we get x 2y + 5 = 0 3. We introduce a parameter by letting y = t. Then by 3 we get x = 5 + 2t. By 2 we have z = 6 x y = 6 ( 5 + 2t) t = 11 3t. So the parametric equation of the line is The vector equation of the line is x = 5 + 2t y = 0 + t z = 11 3t. [x, y, z] = [ 5, 0, 11] + t[2, 1, 3]. Question: Is the Two planes π 1 and π 2 are perpendicular? why? 15
16 b) π 3 : 2x 6y + 4z 7 = 0 1 π 4 : 3x 9y + 6z 2 = 0 2 c) π 5 : x + y 2z + 2 = 0 1 π 6 : 2x + 2y 4z + 4 =
17 Intersection of Three Planes in Three-Space Geometrically there are eight possibilities for the intersection of two planes in three-space. 1 Interest at a point 2 Intersect in a line Note about the normals: 3 Three planes are coincident. 4 Two planes are coincident and the third plane is not parallel. Note about the normals: 5 Three planes are parallel. 6 Two planes are coincident and the third plane is parallel to the first two planes. 17
18 Note about the normals: 7 Two planes are parallel and the third plane is not parallel. 8 Pairs of planes intersect in lines that are parallel. Note about the normals: Theorem Three normal vectors n 1, n 2 and n 3 are coplanar if n 1 n 2 n 3 = 0 Why? what cases does this cover? Algebraically there are three possibilities for the solutions of the following system of equations. a 1 x + b 1 y + c 1 z + d 1 = 0 a 2 x + b 2 y + c 2 z + d 2 = 0 a 3 x + b 3 y + c 3 z + d 3 = 0 18
19 Unique solution (1) Note about the normals: Infinitely many solutions (2-4) Note about the normals: No solution (5-8) Note about the normals: Example 8.18 (Example 2 page 486). For each set of planes, describe the number of solutions and how the planes intersect. a) π 1 : x 5y + 2z 10 = 0 1 π 2 : x + 7y 2z + 6 = 0 2 π 3 : 8x + 5y + z 20 = 0 3 Solution: Discuss normals: Let 2 1, we get 12y 4z + 16 = 0 4. Let ( 8) 1 + 3, we get 45y 15z + 60 = 0 5. Let , we get 0 = 0 0 which is true for all values of y and z. 19
20 ** = ** ( 2 1 ) (( 8) ) = 0 which implies 3 = 15 We introduce a parameter by letting z = t. Then y = t by 4 or 5. By 1, 3 3 we get x = 5y 2z + 10 = 5( t) 2t + 10 = 1 t. So the parametric equation of the line is The vector equation of the line is b) [x, y, z] = x = t y = t z = 0 + t. [ ] 10 3, 4 3, 0 + t [ 13, 13 ], 1. π 4 : 2x + y + 6z 7 = 0 1 π 5 : 3x + 4y + 3z + 8 = 0 2 π 6 : x 2y 4z 9 =
8. Find r a! r b. a) r a = [3, 2, 7], r b = [ 1, 4, 5] b) r a = [ 5, 6, 7], r b = [2, 7, 4]
Chapter 8 Prerequisite Skills BLM 8-1.. Linear Relations 1. Make a table of values and graph each linear function a) y = 2x b) y = x + 5 c) 2x + 6y = 12 d) x + 7y = 21 2. Find the x- and y-intercepts of
More informationMatrices. A matrix is a method of writing a set of numbers using rows and columns. Cells in a matrix can be referenced in the form.
Matrices A matrix is a method of writing a set of numbers using rows and columns. 1 2 3 4 3 2 1 5 7 2 5 4 2 0 5 10 12 8 4 9 25 30 1 1 Reading Information from a Matrix Cells in a matrix can be referenced
More information12.5 Equations of Lines and Planes
12.5 Equations of Lines and Planes Equation of Lines Vector Equation of Lines Parametric Equation of Lines Symmetric Equation of Lines Relation Between Two Lines Equations of Planes Vector Equation of
More informationLinear Algebra: Homework 3
Linear Algebra: Homework 3 Alvin Lin August 206 - December 206 Section.2 Exercise 48 Find all values of the scalar k for which the two vectors are orthogonal. [ ] [ ] 2 k + u v 3 k u v 0 2(k + ) + 3(k
More informationMATH 1210 Assignment 3 Solutions 17R-T2
MATH 1210 Assignment 3 Solutions 17R-T2 This assignment is optional and does not need to be handed in. Attempt all questions, write out nicely written solutions (showing all your work), and the solutions
More informationSection 8.1 Vector and Parametric Equations of a Line in
Section 8.1 Vector and Parametric Equations of a Line in R 2 In this section, we begin with a discussion about how to find the vector and parametric equations of a line in R 2. To find the vector and parametric
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 2/13/13, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.2. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241sp13/241.html)
More informationVECTORS AND THE GEOMETRY OF SPACE
VECTORS AND THE GEOMETRY OF SPACE VECTORS AND THE GEOMETRY OF SPACE A line in the xy-plane is determined when a point on the line and the direction of the line (its slope or angle of inclination) are given.
More informationWhat you will learn today
What you will learn today The Dot Product Equations of Vectors and the Geometry of Space 1/29 Direction angles and Direction cosines Projections Definitions: 1. a : a 1, a 2, a 3, b : b 1, b 2, b 3, a
More informationVector equations of lines in the plane and 3-space (uses vector addition & scalar multiplication).
Boise State Math 275 (Ultman) Worksheet 1.6: Lines and Planes From the Toolbox (what you need from previous classes) Plotting points, sketching vectors. Be able to find the component form a vector given
More informationMATH 12 CLASS 4 NOTES, SEP
MATH 12 CLASS 4 NOTES, SEP 28 2011 Contents 1. Lines in R 3 1 2. Intersections of lines in R 3 2 3. The equation of a plane 4 4. Various problems with planes 5 4.1. Intersection of planes with planes or
More informationKevin James. MTHSC 206 Section 12.5 Equations of Lines and Planes
MTHSC 206 Section 12.5 Equations of Lines and Planes Definition A line in R 3 can be described by a point and a direction vector. Given the point r 0 and the direction vector v. Any point r on the line
More informationUnit 2: Lines and Planes in 3 Space. Linear Combinations of Vectors
Lesson10.notebook November 28, 2012 Unit 2: Lines and Planes in 3 Space Linear Combinations of Vectors Today's goal: I can write vectors as linear combinations of each other using the appropriate method
More information4.3 Equations in 3-space
4.3 Equations in 3-space istance can be used to define functions from a 3-space R 3 to the line R. Let P be a fixed point in the 3-space R 3 (say, with coordinates P (2, 5, 7)). Consider a function f :
More informationSystems of Linear Equations
Systems of Linear Equations Linear Algebra MATH 2076 Linear Algebra SLEs Chapter 1 Section 1 1 / 8 Linear Equations and their Solutions A linear equation in unknowns (the variables) x 1, x 2,..., x n has
More informationMTH MTH Lecture 6. Yevgeniy Kovchegov Oregon State University
MTH 306 0 MTH 306 - Lecture 6 Yevgeniy Kovchegov Oregon State University MTH 306 1 Topics Lines and planes. Systems of linear equations. Systematic elimination of unknowns. Coe cient matrix. Augmented
More informationMAT1035 Analytic Geometry
MAT1035 Analytic Geometry Lecture Notes R.A. Sabri Kaan Gürbüzer Dokuz Eylül University 2016 2 Contents 1 Review of Trigonometry 5 2 Polar Coordinates 7 3 Vectors in R n 9 3.1 Located Vectors..............................................
More informationMath 2433 Notes Week The Dot Product. The angle between two vectors is found with this formula: cosθ = a b
Math 2433 Notes Week 2 11.3 The Dot Product The angle between two vectors is found with this formula: cosθ = a b a b 3) Given, a = 4i + 4j, b = i - 2j + 3k, c = 2i + 2k Find the angle between a and c Projection
More informationCreated by T. Madas VECTOR PRACTICE Part B Created by T. Madas
VECTOR PRACTICE Part B THE CROSS PRODUCT Question 1 Find in each of the following cases a) a = 2i + 5j + k and b = 3i j b) a = i + 2j + k and b = 3i j k c) a = 3i j 2k and b = i + 3j + k d) a = 7i + j
More informationCulminating Review for Vectors
Culminating Review for Vectors 0011 0010 1010 1101 0001 0100 1011 An Introduction to Vectors Applications of Vectors Equations of Lines and Planes 4 12 Relationships between Points, Lines and Planes An
More informationFINDING THE INTERSECTION OF TWO LINES
FINDING THE INTERSECTION OF TWO LINES REALTIONSHIP BETWEEN LINES 2 D: D: the lines are coplanar (they lie in the same plane). They could be: intersecting parallel coincident the lines are not coplanar
More informationChapter 13: Vectors and the Geometry of Space
Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic
More informationChapter 13: Vectors and the Geometry of Space
Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic
More informationand Calculus and Vectors
and Calculus and Vectors Autograph is spectacular dynamic software from the UK that allows teachers to visualise many of the mathematical topics that occur in the Ontario Grade 12 CALCULUS and VECTORS
More informationAnalytic Geometry MAT 1035
Analytic Geometry MAT 035 5.09.04 WEEKLY PROGRAM - The first week of the semester, we will introduce the course and given a brief outline. We continue with vectors in R n and some operations including
More informationDetailed objectives are given in each of the sections listed below. 1. Cartesian Space Coordinates. 2. Displacements, Forces, Velocities and Vectors
Unit 1 Vectors In this unit, we introduce vectors, vector operations, and equations of lines and planes. Note: Unit 1 is based on Chapter 12 of the textbook, Salas and Hille s Calculus: Several Variables,
More informationCharacteristics of Linear Functions (pp. 1 of 8)
Characteristics of Linear Functions (pp. 1 of 8) Algebra 2 Parent Function Table Linear Parent Function: x y y = Domain: Range: What patterns do you observe in the table and graph of the linear parent
More informationAnalytic Geometry MAT 1035
Analytic Geometry MAT 035 5.09.04 WEEKLY PROGRAM - The first week of the semester, we will introduce the course and given a brief outline. We continue with vectors in R n and some operations including
More informationMAT 1339-S14 Class 8
MAT 1339-S14 Class 8 July 28, 2014 Contents 7.2 Review Dot Product........................... 2 7.3 Applications of the Dot Product..................... 4 7.4 Vectors in Three-Space.........................
More informationLB 220 Homework 4 Solutions
LB 220 Homework 4 Solutions Section 11.4, # 40: This problem was solved in class on Feb. 03. Section 11.4, # 42: This problem was also solved in class on Feb. 03. Section 11.4, # 43: Also solved in class
More informationSection 8.4 Vector and Parametric Equations of a Plane
Section 8.4 Vector and Parametric Equations of a Plane In the previous section, the vector, parametric, and symmetric equations of lines in R 3 were developed. In this section, we will develop vector and
More informationCurriculum Correlation
Curriculum Correlation Ontario Grade 12(MCV4U) Curriculum Correlation Rate of Change Chapter/Lesson/Feature Overall Expectations demonstrate an understanding of rate of change by making connections between
More informationChapter 2 - Vector Algebra
A spatial vector, or simply vector, is a concept characterized by a magnitude and a direction, and which sums with other vectors according to the Parallelogram Law. A vector can be thought of as an arrow
More informationSections 8.1 & 8.2 Systems of Linear Equations in Two Variables
Sections 8.1 & 8.2 Systems of Linear Equations in Two Variables Department of Mathematics Porterville College September 7, 2014 Systems of Linear Equations in Two Variables Learning Objectives: Solve Systems
More informationThree-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems
To locate a point in a plane, two numbers are necessary. We know that any point in the plane can be represented as an ordered pair (a, b) of real numbers, where a is the x-coordinate and b is the y-coordinate.
More information5. A triangle has sides represented by the vectors (1, 2) and (5, 6). Determine the vector representing the third side.
Vectors EXAM review Problem 1 = 8 and = 1 a) Find the net force, assume that points North, and points East b) Find the equilibrant force 2 = 15, = 7, and the angle between and is 60 What is the magnitude
More informationChapter 1: Precalculus Review
: Precalculus Review Math 115 17 January 2018 Overview 1 Important Notation 2 Exponents 3 Polynomials 4 Rational Functions 5 Cartesian Coordinates 6 Lines Notation Intervals: Interval Notation (a, b) (a,
More informationSection 1.1 System of Linear Equations. Dr. Abdulla Eid. College of Science. MATHS 211: Linear Algebra
Section 1.1 System of Linear Equations College of Science MATHS 211: Linear Algebra (University of Bahrain) Linear System 1 / 33 Goals:. 1 Define system of linear equations and their solutions. 2 To represent
More informationVECTORS IN COMPONENT FORM
VECTORS IN COMPONENT FORM In Cartesian coordinates any D vector a can be written as a = a x i + a y j + a z k a x a y a x a y a z a z where i, j and k are unit vectors in x, y and z directions. i = j =
More informationchapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true?
chapter vector geometry solutions V. Exercise A. For the shape shown, find a single vector which is equal to a)!!! " AB + BC AC b)! AD!!! " + DB AB c)! AC + CD AD d)! BC + CD!!! " + DA BA e) CD!!! " "
More information(1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3
Math 127 Introduction and Review (1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3 MATH 127 Introduction to Calculus III
More informationDistances in R 3. Last time we figured out the (parametric) equation of a line and the (scalar) equation of a plane:
Distances in R 3 Last time we figured out the (parametric) equation of a line and the (scalar) equation of a plane: Definition: The equation of a line through point P(x 0, y 0, z 0 ) with directional vector
More informationThe Circle. 1 The equation of a circle. Nikos Apostolakis. Spring 2017
The Circle Nikos Apostolakis Spring 017 1 The equation of a circle Definition 1. Given a point C in the plane and a positive number r, the circle with center C and radius r, is the locus of points that
More informationGraphing Systems of Linear Equations
Graphing Systems of Linear Equations Groups of equations, called systems, serve as a model for a wide variety of applications in science and business. In these notes, we will be concerned only with groups
More informationWe have seen that for a function the partial derivatives whenever they exist, play an important role. This motivates the following definition.
\ Module 12 : Total differential, Tangent planes and normals Lecture 34 : Gradient of a scaler field [Section 34.1] Objectives In this section you will learn the following : The notions gradient vector
More informationSKILL BUILDER TEN. Graphs of Linear Equations with Two Variables. If x = 2 then y = = = 7 and (2, 7) is a solution.
SKILL BUILDER TEN Graphs of Linear Equations with Two Variables A first degree equation is called a linear equation, since its graph is a straight line. In a linear equation, each term is a constant or
More informationIntro Vectors 2D implicit curves 2D parametric curves. Graphics 2011/2012, 4th quarter. Lecture 2: vectors, curves, and surfaces
Lecture 2, curves, and surfaces Organizational remarks Tutorials: Tutorial 1 will be online later today TA sessions for questions start next week Practicals: Exams: Make sure to find a team partner very
More informationGraphical Solutions of Linear Systems
Graphical Solutions of Linear Systems Consistent System (At least one solution) Inconsistent System (No Solution) Independent (One solution) Dependent (Infinite many solutions) Parallel Lines Equations
More informationThe Sphere OPTIONAL - I Vectors and three dimensional Geometry THE SPHERE
36 THE SPHERE You must have played or seen students playing football, basketball or table tennis. Football, basketball, table tennis ball are all examples of geometrical figures which we call "spheres"
More informationLinear Equations in Two Variables
Linear Equations in Two Variables In this chapter, we ll use the geometry of lines to help us solve equations. Linear equations in two variables. If a, b, andr are real numbers (and if a and b are not
More informationLinear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}
Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in
More informationLesson 3. Perpendicularity, Planes, and Cross Products
Lesson 3 Perpendicularity, Planes, and Cross Products Example 1: Equation for a Plane Let P (2,3, 1) be a point in space and let V (4, 2,5) be a vector. Find the xyz-equation of the plane containing P
More informationHow can we find the distance between a point and a plane in R 3? Between two lines in R 3? Between two planes? Between a plane and a line?
Overview Yesterday we introduced equations to describe lines and planes in R 3 : r = r 0 + tv The vector equation for a line describes arbitrary points r in terms of a specific point r 0 and the direction
More informationVectors. 1 Basic Definitions. Liming Pang
Vectors Liming Pang 1 Basic Definitions Definition 1. A vector in a line/plane/space is a quantity which has both magnitude and direction. The magnitude is a nonnegative real number and the direction is
More informationPrecalculus: Linear Equations Practice Problems. Questions. 1. Solve for x when 2 3 x = 1 15 x Solve for x when x 2 + x 5 = 7 10.
Questions. Solve for x when 3 x = 5 x + 3 5.. Solve for x when x + x 5 = 7 0. 3. Solve for x when 0 3 x = x. 4. Is 4 a solution to (y ) + = 3 (3y 4)? 8 5. Solve for x when 4 5 x 3 = 3x +. 6. Solve for
More informationMultiple forces or velocities influencing an object, add as vectors.
September 23, 2018 Coming up: Mon 10/1: Exploration Topic Due! Wed 10/10: PSAT Fri 10/12: Vector Unit Exam (Ch 12 & 13) Fri 10/12: Begin Exploration writing Wed 10/31: Exploration Final Due! 1. Apply vector
More informationIntro Vectors 2D implicit curves 2D parametric curves. Graphics 2012/2013, 4th quarter. Lecture 2: vectors, curves, and surfaces
Lecture 2, curves, and surfaces Organizational remarks Tutorials: TA sessions for tutorial 1 start today Tutorial 2 will go online after lecture 3 Practicals: Make sure to find a team partner very soon
More informationMathematics Syllabus UNIT I ALGEBRA : 1. SETS, RELATIONS AND FUNCTIONS
Mathematics Syllabus UNIT I ALGEBRA : 1. SETS, RELATIONS AND FUNCTIONS (i) Sets and their Representations: Finite and Infinite sets; Empty set; Equal sets; Subsets; Power set; Universal set; Venn Diagrams;
More informationSOLUTIONS FOR PROBLEMS 1-30
. Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).
More information6-4 Solving Special Systems
6-4 Solving Special Systems Warm Up Lesson Presentation Lesson Quiz 1 2 pts Bell Quiz 6-4 Solve the equation. 1. 2(x + 1) = 2x + 2 3 pts Solve by using any method. 2. y = 3x + 2 2x + y = 7 5 pts possible
More informationDistance Formula in 3-D Given any two points P 1 (x 1, y 1, z 1 ) and P 2 (x 2, y 2, z 2 ) the distance between them is ( ) ( ) ( )
Vectors and the Geometry of Space Vector Space The 3-D coordinate system (rectangular coordinates ) is the intersection of three perpendicular (orthogonal) lines called coordinate axis: x, y, and z. Their
More information11.1 Three-Dimensional Coordinate System
11.1 Three-Dimensional Coordinate System In three dimensions, a point has three coordinates: (x,y,z). The normal orientation of the x, y, and z-axes is shown below. The three axes divide the region into
More informationMATH 12 CLASS 2 NOTES, SEP Contents. 2. Dot product: determining the angle between two vectors 2
MATH 12 CLASS 2 NOTES, SEP 23 2011 Contents 1. Dot product: definition, basic properties 1 2. Dot product: determining the angle between two vectors 2 Quick links to definitions/theorems Dot product definition
More information6.1.1 Angle between Two Lines Intersection of Two lines Shortest Distance from a Point to a Line
CHAPTER 6 : VECTORS 6. Lines in Space 6.. Angle between Two Lines 6.. Intersection of Two lines 6..3 Shortest Distance from a Point to a Line 6. Planes in Space 6.. Intersection of Two Planes 6.. Angle
More informationMathematics 308 Geometry. Chapter 2. Elementary coordinate geometry
Mathematics 308 Geometry Chapter 2. Elementary coordinate geometry Using a computer to produce pictures requires translating geometry to numbers, which is carried out through a coordinate system. Through
More information12.1. Cartesian Space
12.1. Cartesian Space In most of your previous math classes, we worked with functions on the xy-plane only meaning we were working only in 2D. Now we will be working in space, or rather 3D. Now we will
More informationVectors. Section 3: Using the vector product
Vectors Section 3: Using the vector product Notes and Examples These notes contain subsections on Using the vector product in finding the equation of a plane The intersection of two planes The distance
More informationAlgebra 2 Prep. Name Period
Algebra 2 Prep Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing
More informationCalculus and Vectors, Grade 12
Calculus and Vectors, Grade University Preparation MCV4U This course builds on students previous eperience with functions and their developing understanding of rates of change. Students will solve problems
More informationBasic Surveying Week 3, Lesson 2 Semester 2017/18/2 Vectors, equation of line, circle, ellipse
Basic Surveying Week 3, Lesson Semester 017/18/ Vectors, equation of line, circle, ellipse 1. Introduction In surveying calculations, we use the two or three dimensional coordinates of points or objects
More information(iii) converting between scalar product and parametric forms. (ii) vector perpendicular to two given (3D) vectors
Vector Theory (15/3/2014) www.alevelmathsng.co.uk Contents (1) Equation of a line (i) parametric form (ii) relation to Cartesian form (iii) vector product form (2) Equation of a plane (i) scalar product
More informationChapter 5. Linear Algebra. Sections A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra Sections 5.1 5.3 A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are
More informationEuclidean Spaces. Euclidean Spaces. Chapter 10 -S&B
Chapter 10 -S&B The Real Line: every real number is represented by exactly one point on the line. The plane (i.e., consumption bundles): Pairs of numbers have a geometric representation Cartesian plane
More informationPut the following equations to slope-intercept form then use 2 points to graph
Tuesday September 23, 2014 Warm-up: Put the following equations to slope-intercept form then use 2 points to graph 1. 4x - 3y = 8 8 x 6y = 16 2. 2x + y = 4 2x + y = 1 Tuesday September 23, 2014 Warm-up:
More informationIntroduction. Chapter Points, Vectors and Coordinate Systems
Chapter 1 Introduction Computer aided geometric design (CAGD) concerns itself with the mathematical description of shape for use in computer graphics, manufacturing, or analysis. It draws upon the fields
More informationLecture 1: Systems of linear equations and their solutions
Lecture 1: Systems of linear equations and their solutions Course overview Topics to be covered this semester: Systems of linear equations and Gaussian elimination: Solving linear equations and applications
More informationMath 51, Homework-2. Section numbers are from the course textbook.
SSEA Summer 2017 Math 51, Homework-2 Section numbers are from the course textbook. 1. Write the parametric equation of the plane that contains the following point and line: 1 1 1 3 2, 4 2 + t 3 0 t R.
More informationHerndon High School Geometry Honors Summer Assignment
Welcome to Geometry! This summer packet is for all students enrolled in Geometry Honors at Herndon High School for Fall 07. The packet contains prerequisite skills that you will need to be successful in
More informationUnit 8. ANALYTIC GEOMETRY.
Unit 8. ANALYTIC GEOMETRY. 1. VECTORS IN THE PLANE A vector is a line segment running from point A (tail) to point B (head). 1.1 DIRECTION OF A VECTOR The direction of a vector is the direction of the
More informationChapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1
More informationLecture 5. Equations of Lines and Planes. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.
Lecture 5 Equations of Lines and Planes Dan Nichols nichols@math.umass.edu MATH 233, Spring 2018 Universit of Massachusetts Februar 6, 2018 (2) Upcoming midterm eam First midterm: Wednesda Feb. 21, 7:00-9:00
More informationAnnouncements. Topics: Homework: - section 12.4 (cross/vector product) - section 12.5 (equations of lines and planes)
Topics: Announcements - section 12.4 (cross/vector product) - section 12.5 (equations of lines and planes) Homework: ü review lecture notes thoroughl ü work on eercises from the tetbook in sections 12.4
More informationSHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Sketch the region bounded between the given curves and then find the area of the region. ) y =, y = )
More informationMidterm 1 Review. Distance = (x 1 x 0 ) 2 + (y 1 y 0 ) 2.
Midterm 1 Review Comments about the midterm The midterm will consist of five questions and will test on material from the first seven lectures the material given below. No calculus either single variable
More informationCHAPTER 1 Systems of Linear Equations
CHAPTER Systems of Linear Equations Section. Introduction to Systems of Linear Equations. Because the equation is in the form a x a y b, it is linear in the variables x and y. 0. Because the equation cannot
More information( 7, 3) means x = 7 and y = 3. ( 7, 3) works in both equations so. Section 5 1: Solving a System of Linear Equations by Graphing
Section 5 : Solving a Sstem of Linear Equations b Graphing What is a sstem of Linear Equations? A sstem of linear equations is a list of two or more linear equations that each represents the graph of a
More informationTS EAMCET 2016 SYLLABUS ENGINEERING STREAM
TS EAMCET 2016 SYLLABUS ENGINEERING STREAM Subject: MATHEMATICS 1) ALGEBRA : a) Functions: Types of functions Definitions - Inverse functions and Theorems - Domain, Range, Inverse of real valued functions.
More informationFunction Junction: Homework Examples from ACE
Function Junction: Homework Examples from ACE Investigation 1: The Families of Functions, ACE #5, #10 Investigation 2: Arithmetic and Geometric Sequences, ACE #4, #17 Investigation 3: Transforming Graphs,
More information1.2 Graphs and Lines. Cartesian Coordinate System
1.2 Graphs and Lines Cartesian Coordinate System Note that there is a one-to-one correspondence between the points in a plane and the elements in the set of all ordered pairs (a, b) of real numbers. Graphs
More informationReteach 2-3. Graphing Linear Functions. 22 Holt Algebra 2. Name Date Class
-3 Graphing Linear Functions Use intercepts to sketch the graph of the function 3x 6y 1. The x-intercept is where the graph crosses the x-axis. To find the x-intercept, set y 0 and solve for x. 3x 6y 1
More informationIntroduction to Vectors
Introduction to Vectors K. Behrend January 31, 008 Abstract An introduction to vectors in R and R 3. Lines and planes in R 3. Linear dependence. 1 Contents Introduction 3 1 Vectors 4 1.1 Plane vectors...............................
More informationIntroduction to Linear Algebra, Second Edition, Serge Lange
Introduction to Linear Algebra, Second Edition, Serge Lange Chapter I: Vectors R n defined. Addition and scalar multiplication in R n. Two geometric interpretations for a vector: point and displacement.
More informationChapter 5. Linear Algebra. Sections A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra Sections 5.1 5.3 A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are
More informationMini Lecture 2.1 Introduction to Functions
Mini Lecture.1 Introduction to Functions 1. Find the domain and range of a relation.. Determine whether a relation is a function. 3. Evaluate a function. 1. Find the domain and range of the relation. a.
More informationLinear algebra. NEU 466M Instructor: Professor Ila R. Fiete Spring 2016
Linear algebra NEU M Instructor: Professor Ila R. Fiete Spring 01 NotaBon Matrices: upper-case A, B, U, W Vector: bold, (usually) lower-case x, y, v, w x! x (handwribng: ) Elements of matrix, vector: lower-case
More informationGraphing Linear Systems
Graphing Linear Systems Goal Estimate the solution of a system of linear equations by graphing. VOCABULARY System of linear equations A system of linear equations is two or more linear equations in the
More informationGraphing Linear Equations and Inequalities: Proficiency Exam
Connexions module: m22015 1 Graphing Linear Equations and Inequalities: Proficiency Exam Denny Burzynski Wade Ellis This work is produced by The Connexions Project and licensed under the Creative Commons
More informationPOINTS, LINES, DISTANCES
POINTS, LINES, DISTANCES NIKOS APOSTOLAKIS Examples/Exercises: (1) Find the equation of the line that passes through (4, 5), (4, ) () Find the equation of the line that passes through the points (1, ),
More information1.1 Single Variable Calculus versus Multivariable Calculus Rectangular Coordinate Systems... 4
MATH2202 Notebook 1 Fall 2015/2016 prepared by Professor Jenny Baglivo Contents 1 MATH2202 Notebook 1 3 1.1 Single Variable Calculus versus Multivariable Calculus................... 3 1.2 Rectangular Coordinate
More informationStudy guide for Exam 1. by William H. Meeks III October 26, 2012
Study guide for Exam 1. by William H. Meeks III October 2, 2012 1 Basics. First we cover the basic definitions and then we go over related problems. Note that the material for the actual midterm may include
More information