Vectors Coordinate frames 2D implicit curves 2D parametric curves. Graphics 2008/2009, period 1. Lecture 2: vectors, curves, and surfaces

Size: px
Start display at page:

Download "Vectors Coordinate frames 2D implicit curves 2D parametric curves. Graphics 2008/2009, period 1. Lecture 2: vectors, curves, and surfaces"

Transcription

1 Graphics 2008/2009, period 1 Lecture 2 Vectors, curves, and surfaces

2 Computer graphics example: Pixar (source:

3 Computer graphics example: Pixar (source:

4 Computer graphics example: Pixar (source:

5 Computer graphics example: Pixar (source:

6 Computer graphics example: Pixar (source:

7 Computer graphics example: Pixar (source:

8 Computer graphics example: Pixar (source:

9 Computer graphics example: Pixar (source:

10 Computer graphics example: Pixar (source:

11 Computer graphics example: Pixar (source:

12 Computer graphics example: Pixar (source:

13 Computer graphics example: Pixar (source:

14 Computer graphics example: Pixar (source:

15 Computer graphics example: Pixar (source:

16 Computer graphics example: Pixar (source:

17 Computer graphics example: Pixar (source:

18 Computer graphics example: Pixar Modeling Lighting Animating (source: Shading

19 Recap Computer graphics: Modelling (creating 3d virtual worlds) Rendering (creating shaded 2d images from 3d models) Two basic methods: Projective methods Ray tracing Last time: Lines/rays and spheres Intersections between them Today: More mathematics to describe objects

20 : definition Vectors Length Addition Bases Dot product and cross product A vector is specified by a length and direction In R d it can be defined as v 1 v 2 an ordered d-tuple v =.. v d v ( ) 5 v = 1

21 : definition Vectors Length Addition Bases Dot product and cross product v 1 In R 3, for example: v = v 2 v 3 v x x v or v y, or y v, v z z v or (v 1, v 2, v 3 ), or (v 1, v 2, v 3 ) T, or... v ( ) 5 v = 1

22 Vectors: algebraic interpretation Length Addition Bases Dot product and cross product A 2D vector (x v, y v ) can be seen as the point (x v, y v ) in the Cartesian plane. v ( ) 5 v = 1

23 Vectors: geometric interpretation Length Addition Bases Dot product and cross product A 2D vector (x v, y v ) can be seen as an offset from the origin. Such an offset (arrow) can be translated. v v v v = ( ) 5 1

24 : length Vectors Length Addition Bases Dot product and cross product The Euclidean length of a d-dimensional vector v is v = v1 2 + v v2 d v ( ) 5 v = 1

25 Vectors: scalar multiple Length Addition Bases Dot product and cross product A scalar multiple of a d-dimensional vector v is λv = (λv 1, λv 2,..., λv d ) Note that v and λv have the same direction or opposite directions

26 Parallel vectors Vectors Length Addition Bases Dot product and cross product Two vectors v 1 and v 2 are parallel if one is a scalar multiple of the other, i.e., there is a λ such that v 2 = λv 1. Note that if one of the vectors is the null vector, then the vectors are considered neither parallel nor not parallel.

27 Unit vectors Vectors Length Addition Bases Dot product and cross product A vector v is a unit vector if v = 1. Normalization Q: given an arbitrary vector v, how do we find a unit vector parallel to v? Can every vector be normalized?

28 Addition of vectors Vectors Length Addition Bases Dot product and cross product Given two vectors in R d, v = (v 1, v 2,..., v d ) and w = (w 1, w 2,..., w d ) their sum is defined as v + w = (v 1 + w 1, v 2 + w 2,..., v d + w d )

29 Addition of vectors Vectors Length Addition Bases Dot product and cross product Addition of vectors is commutative, as can be seen easily from the geometric interpretation. Q: show algebraically that vector addition is commutative. Q: what is the relation between v, w, and v + w?

30 Addition of vectors Vectors Length Addition Bases Dot product and cross product Q: How would subtraction be defined?

31 Bases Vectors Length Addition Bases Dot product and cross product A 2D vector can be expressed as a combination of any pair of non-parallel vectors. For instance, in the image, a = 1.5v + 0.6w. Such a pair is called linearly independent, and they form a 2D basis. The extension to higher dimensions is straightforward. v w a

32 Orthonormal basis Vectors Length Addition Bases Dot product and cross product Two vectors form an orthonormal basis if (1) they are orthogonal to each other, and (2) they are unit vectors. The advantage of an orthonormal basis is that lengths of vectors, expressed in the basis, are easy to calculate.

33 Length Addition Bases Dot product and cross product Dot product (inner product, scalar product) For two vectors v, w R d, the dot product is defined as v w = v 1 w 1 +v 2 w 2 + +v d w d, or v w = d i=1 v iw i We have that cos θ = v w v w, where θ is the angle between the two vectors. dot product

34 Length Addition Bases Dot product and cross product Dot product (inner product, scalar product) dot product

35 Length Addition Bases Dot product and cross product Dot product (inner product, scalar product) Q: what is the inner product of an arbitrary unit vector with itself? Q: what do we know if for two vectors v and w we have that v w = 0? dot product

36 Length Addition Bases Dot product and cross product Dot product (inner product, scalar product) dot product

37 Length Addition Bases Dot product and cross product Dot product (inner product, scalar product) Another use: Calculate the length of the projection of one vector onto another. dot product

38 Cross product Vectors Length Addition Bases Dot product and cross product For two vectors v, w R 3, the cross product is defined as v 2 w 3 v 3 w 2 v w = v 3 w 1 v 1 w 3 v 1 w 2 v 2 w 1 cross product We have that v w = v w sin θ, where θ is the angle between v and w.

39 Cross product Vectors Length Addition Bases Dot product and cross product Q: show that v w is orthogonal to both v and w. cross product

40 Cross product Q: what is v v? Vectors Length Addition Bases Dot product and cross product Q: what is v w if v and w are parallel? cross product

41 Cross product Vectors Length Addition Bases Dot product and cross product Q: is it possible or necessary that v and w are orthogonal to form v w? cross product

42 Left- and right-handed systems Bases and coordinate frames Left- and right-handed systems Coordinate systems in 3D come in two flavors: left-handed and right-handed. There are arguments for both left- and right-handed systems for The global system The camera system Object systems Z X Y Y Y Z X X Z Camera coordinate system Z Y Y X X Z World coordinate system

43 Left- and right-handed systems Bases and coordinate frames Coordinate transformations A frequent operation in graphics is the change from one coordinate system (e.g., the (u, v, w) camera system) to another (e.g., the (x, y, z) global system). v u w Having orthonormal bases for both systems makes the transformations simpler. Z Y X

44 General curves Circles Lines 2D Implicit curves f(x, y) y An implicit curve in 2D has the form x f(x, y) = 0 f maps two-dimensional points to a real value; the points for which this value is 0 are on the curve, while other points are not on the curve. y f(x, y) = 0 x

45 General curves Circles Lines Implicit representation of circles The implicit representation of a 2D circle with center c and radius r is p (x x c ) 2 + (y y c ) 2 r 2 = 0 c So for any point p that lies on the circle, we have that (p c) (p c) r 2 = 0, so p c 2 r 2 = 0, which gives p c = r.

46 General curves Circles Lines Implicit representation of circles The implicit representation of a 2D circle with center c and radius r is (x x c ) 2 + (y y c ) 2 r 2 = 0 So for any point p that lies on the circle, we have that (p c) (p c) r 2 = 0, so p c 2 r 2 = 0, which gives p c = r.

47 General curves Circles Lines Implicit representation of lines A well-known representation of lines is the slope-intercept form y = ax + b This can easily be converted to ax + y b = 0. If b = 0, the line intersects the origin, and we have ( ) x n p = 0, with p = and y ( ) a n = 1

48 General curves Circles Lines Implicit representation of lines n p = 0, with p = ( ) x, n = y ( ) a 1 Q: what if the line does not intersect the origin?

49 General curves Circles Lines Implicit representation of lines ( ) a Note: The vector n = 1 is orthogonal to the line. (This is true for the first 2 coefficiants of any implicit line representation.)

50 General curves Circles Lines Implicit representation of lines

51 General curves Circles Lines parametric implicit representation A parametric curve is controlled by a single parameter, and has the form ( ) x = y ( ) g(t) h(t) Parametric representations have some advantages over functions, even if a function would suffice to represent the curve. h(t) g(t)

52 General curves Circles Lines parametric implicit representation Parametric equation of a circle The parametric equation of a 2D circle with center c and radius r is ( ) x y = ( ) xc + r cos φ y c + r sin φ p φ c

53 General curves Circles Lines parametric implicit representation Parametric equation of a line The parametric equation of a line through the points p 0 and p 1 is ( ) x = y ( ) xp0 + t(x p1 x p0 ) y p0 + t(y p1 y p0 ) As we have seen before, this can alternatively be written as p(t) = p 0 + t(p 1 p 0 ). p 0 p 1

54 General curves Circles Lines parametric implicit representation Conversion between representations It is convenient to be able to convert a parametric equation of a line into an implicit equation, and vice versa. Q: how do we do that? p 0 p 1

55 General curves Circles Lines parametric implicit representation Conversion between representations Q: how do we do that? Slope-intercept repr.: y = ax + b Implicit representation: f(x, y) = ax + by + c = 0. Parametric representation (vector form): p(t) = p 0 + t(p 1 p 0 ) (Note: These are not the same a,b s for the different equations!) p 0 p 1

Intro Vectors 2D implicit curves 2D parametric curves. Graphics 2011/2012, 4th quarter. Lecture 2: vectors, curves, and surfaces

Intro 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 information

Intro Vectors 2D implicit curves 2D parametric curves. Graphics 2012/2013, 4th quarter. Lecture 2: vectors, curves, and surfaces

Intro 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 information

Elementary maths for GMT

Elementary maths for GMT Elementary maths for GMT Linear Algebra Part 1: Vectors, Representations Algebra and Linear Algebra Algebra: numbers and operations on numbers 2 + 3 = 5 3 7 = 21 Linear Algebra: tuples, triples... of numbers

More information

Euclidean Spaces. Euclidean Spaces. Chapter 10 -S&B

Euclidean 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 information

Computer Graphics MTAT Raimond Tunnel

Computer Graphics MTAT Raimond Tunnel Computer Graphics MTAT.03.015 Raimond Tunnel Points and Vectors In computer graphics we distinguish: Point a location in space (location vector, kohavektor) Vector a direction in space (direction vector,

More information

Study guide for Exam 1. by William H. Meeks III October 26, 2012

Study 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

Lecture 02: 2D Geometry. August 24, 2017

Lecture 02: 2D Geometry. August 24, 2017 Lecture 02: 2D Geometry August 24, 207 This Came Up Tuesday Know what radiosity computation does. Do not expect to implement nor see underlying equations this semester. 8/24/7 CSU CS40 Fall 207, Ross Beveridge

More information

Basic Surveying Week 3, Lesson 2 Semester 2017/18/2 Vectors, equation of line, circle, ellipse

Basic 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

Review of Coordinate Systems

Review of Coordinate Systems Vector in 2 R and 3 R Review of Coordinate Systems Used to describe the position of a point in space Common coordinate systems are: Cartesian Polar Cartesian Coordinate System Also called rectangular coordinate

More information

Math 241, Exam 1 Information.

Math 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 information

Lecture 1: Systems of linear equations and their solutions

Lecture 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 information

Math 302 Outcome Statements Winter 2013

Math 302 Outcome Statements Winter 2013 Math 302 Outcome Statements Winter 2013 1 Rectangular Space Coordinates; Vectors in the Three-Dimensional Space (a) Cartesian coordinates of a point (b) sphere (c) symmetry about a point, a line, and a

More information

Vectors. The standard geometric definition of vector is as something which has direction and magnitude but not position.

Vectors. The standard geometric definition of vector is as something which has direction and magnitude but not position. Vectors The standard geometric definition of vector is as something which has direction and magnitude but not position. Since vectors have no position we may place them wherever is convenient. Vectors

More information

Linear Algebra and Dirac Notation, Pt. 1

Linear Algebra and Dirac Notation, Pt. 1 Linear Algebra and Dirac Notation, Pt. 1 PHYS 500 - Southern Illinois University February 1, 2017 PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 1 February 1, 2017 1 / 13

More information

Chapter 13: Vectors and the Geometry of Space

Chapter 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 information

Chapter 13: Vectors and the Geometry of Space

Chapter 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 information

Mathematical Foundations: Intro

Mathematical Foundations: Intro Mathematical Foundations: Intro Graphics relies on 3 basic objects: 1. Scalars 2. Vectors 3. Points Mathematically defined in terms of spaces: 1. Vector space 2. Affine space 3. Euclidean space Math required:

More information

Graphics (INFOGR), , Block IV, lecture 1 Deb Panja. Today: Vectors, and vector algebra in 2D. Welcome

Graphics (INFOGR), , Block IV, lecture 1 Deb Panja. Today: Vectors, and vector algebra in 2D. Welcome Graphics (INFOGR), 2017-18, Block IV, lecture 1 Deb Panja Today: Vectors, and vector algebra in 2D Welcome 1 Ray tracing part I of the course 2 Ray tracing part I of the course Why vectors? you need to

More information

12.5 Equations of Lines and Planes

12.5 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 information

CS 4204 Computer Graphics

CS 4204 Computer Graphics CS 4204 Computer Graphics Vector and Matrix Yong Cao Virginia Tech Vectors N-tuple: Vectors N-tuple tuple: Magnitude: Unit vectors Normalizing a vector Operations with vectors Addition Multiplication with

More information

Midterm 1 Review. Distance = (x 1 x 0 ) 2 + (y 1 y 0 ) 2.

Midterm 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 information

which arises when we compute the orthogonal projection of a vector y in a subspace with an orthogonal basis. Hence assume that P y = A ij = x j, x i

which arises when we compute the orthogonal projection of a vector y in a subspace with an orthogonal basis. Hence assume that P y = A ij = x j, x i MODULE 6 Topics: Gram-Schmidt orthogonalization process We begin by observing that if the vectors {x j } N are mutually orthogonal in an inner product space V then they are necessarily linearly independent.

More information

Stereographic projection and inverse geometry

Stereographic projection and inverse geometry Stereographic projection and inverse geometry The conformal property of stereographic projections can be established fairly efficiently using the concepts and methods of inverse geometry. This topic is

More information

Department of Mathematical and Statistical Sciences University of Alberta

Department of Mathematical and Statistical Sciences University of Alberta MATH 214 (R1) Winter 2008 Intermediate Calculus I Solutions to Problem Set #8 Completion Date: Friday March 14, 2008 Department of Mathematical and Statistical Sciences University of Alberta Question 1.

More information

Review of Linear Algebra

Review of Linear Algebra Review of Linear Algebra Definitions An m n (read "m by n") matrix, is a rectangular array of entries, where m is the number of rows and n the number of columns. 2 Definitions (Con t) A is square if m=

More information

Linear Algebra Massoud Malek

Linear Algebra Massoud Malek CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product

More information

Introduction. Chapter Points, Vectors and Coordinate Systems

Introduction. 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 information

Distance 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 ( ) ( ) ( )

Distance 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 information

Linear Algebra & Geometry why is linear algebra useful in computer vision?

Linear Algebra & Geometry why is linear algebra useful in computer vision? Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia

More information

MAT1035 Analytic Geometry

MAT1035 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 information

CG Basics 1 of 10. Mathematical Foundations: Vectors, Matrices, & Parametric Equations

CG Basics 1 of 10. Mathematical Foundations: Vectors, Matrices, & Parametric Equations CG Basics 1 of 10 ematical Foundations: Vectors, Matrices, & Parametric Equations William H. Hsu Department of Computing and Information Sciences, KSU KSOL course page: http://bit.ly/hgvxlh Course web

More information

Culminating Review for Vectors

Culminating 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 information

Department of Physics, Korea University

Department of Physics, Korea University Name: Department: Notice +2 ( 1) points per correct (incorrect) answer. No penalty for an unanswered question. Fill the blank ( ) with (8) if the statement is correct (incorrect).!!!: corrections to an

More information

LINEAR ALGEBRA - CHAPTER 1: VECTORS

LINEAR ALGEBRA - CHAPTER 1: VECTORS LINEAR ALGEBRA - CHAPTER 1: VECTORS A game to introduce Linear Algebra In measurement, there are many quantities whose description entirely rely on magnitude, i.e., length, area, volume, mass and temperature.

More information

MATH 255 Applied Honors Calculus III Winter Midterm 1 Review Solutions

MATH 255 Applied Honors Calculus III Winter Midterm 1 Review Solutions MATH 55 Applied Honors Calculus III Winter 11 Midterm 1 Review Solutions 11.1: #19 Particle starts at point ( 1,, traces out a semicircle in the counterclockwise direction, ending at the point (1,. 11.1:

More information

Vectors. 1 Basic Definitions. Liming Pang

Vectors. 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 information

(1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3

(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 information

Course Notes Math 275 Boise State University. Shari Ultman

Course Notes Math 275 Boise State University. Shari Ultman Course Notes Math 275 Boise State University Shari Ultman Fall 2017 Contents 1 Vectors 1 1.1 Introduction to 3-Space & Vectors.............. 3 1.2 Working With Vectors.................... 7 1.3 Introduction

More information

e 2 = e 1 = e 3 = v 1 (v 2 v 3 ) = det(v 1, v 2, v 3 ).

e 2 = e 1 = e 3 = v 1 (v 2 v 3 ) = det(v 1, v 2, v 3 ). 3. Frames In 3D space, a sequence of 3 linearly independent vectors v 1, v 2, v 3 is called a frame, since it gives a coordinate system (a frame of reference). Any vector v can be written as a linear combination

More information

This pre-publication material is for review purposes only. Any typographical or technical errors will be corrected prior to publication.

This pre-publication material is for review purposes only. Any typographical or technical errors will be corrected prior to publication. This pre-publication material is for review purposes only. Any typographical or technical errors will be corrected prior to publication. Copyright Pearson Canada Inc. All rights reserved. Copyright Pearson

More information

Vector equations of lines in the plane and 3-space (uses vector addition & scalar multiplication).

Vector 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 information

Lecture 5 3D polygonal modeling Part 1: Vector graphics Yong-Jin Liu.

Lecture 5 3D polygonal modeling Part 1: Vector graphics Yong-Jin Liu. Fundamentals of Computer Graphics Lecture 5 3D polygonal modeling Part 1: Vector graphics Yong-Jin Liu liuyongjin@tsinghua.edu.cn Material by S.M.Lea (UNC) Introduction In computer graphics, we work with

More information

Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems

Three-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 information

Linear Algebra & Geometry why is linear algebra useful in computer vision?

Linear Algebra & Geometry why is linear algebra useful in computer vision? Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia

More information

The Geometry of R n. Supplemental Lecture Notes for Linear Algebra Courses at Georgia Tech

The Geometry of R n. Supplemental Lecture Notes for Linear Algebra Courses at Georgia Tech The Geometry of R n Supplemental Lecture Notes for Linear Algebra Courses at Georgia Tech Contents Vectors in R n. Vectors....................................... The Length and Direction of a Vector......................3

More information

Linear Algebra I. Ronald van Luijk, 2015

Linear Algebra I. Ronald van Luijk, 2015 Linear Algebra I Ronald van Luijk, 2015 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents Dependencies among sections 3 Chapter 1. Euclidean space: lines and hyperplanes 5 1.1. Definition

More information

I&C 6N. Computational Linear Algebra

I&C 6N. Computational Linear Algebra I&C 6N Computational Linear Algebra 1 Lecture 1: Scalars and Vectors What is a scalar? Computer representation of a scalar Scalar Equality Scalar Operations Addition and Multiplication What is a vector?

More information

Remark 3.2. The cross product only makes sense in R 3.

Remark 3.2. The cross product only makes sense in R 3. 3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with

More information

4.3 Equations in 3-space

4.3 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 information

Exam 1 Review SOLUTIONS

Exam 1 Review SOLUTIONS 1. True or False (and give a short reason): Exam 1 Review SOLUTIONS (a) If the parametric curve x = f(t), y = g(t) satisfies g (1) = 0, then it has a horizontal tangent line when t = 1. FALSE: To make

More information

Lecture 1 Complex Numbers. 1 The field of complex numbers. 1.1 Arithmetic operations. 1.2 Field structure of C. MATH-GA Complex Variables

Lecture 1 Complex Numbers. 1 The field of complex numbers. 1.1 Arithmetic operations. 1.2 Field structure of C. MATH-GA Complex Variables Lecture Complex Numbers MATH-GA 245.00 Complex Variables The field of complex numbers. Arithmetic operations The field C of complex numbers is obtained by adjoining the imaginary unit i to the field R

More information

MAT 1339-S14 Class 8

MAT 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 information

1.1 Single Variable Calculus versus Multivariable Calculus Rectangular Coordinate Systems... 4

1.1 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 information

B553 Lecture 3: Multivariate Calculus and Linear Algebra Review

B553 Lecture 3: Multivariate Calculus and Linear Algebra Review B553 Lecture 3: Multivariate Calculus and Linear Algebra Review Kris Hauser December 30, 2011 We now move from the univariate setting to the multivariate setting, where we will spend the rest of the class.

More information

Vectors and Fields. Vectors versus scalars

Vectors and Fields. Vectors versus scalars C H A P T E R 1 Vectors and Fields Electromagnetics deals with the study of electric and magnetic fields. It is at once apparent that we need to familiarize ourselves with the concept of a field, and in

More information

Linear Algebra Review. Vectors

Linear Algebra Review. Vectors Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors

More information

1.3 LECTURE 3. Vector Product

1.3 LECTURE 3. Vector Product 12 CHAPTER 1. VECTOR ALGEBRA Example. Let L be a line x x 1 a = y y 1 b = z z 1 c passing through a point P 1 and parallel to the vector N = a i +b j +c k. The equation of the plane passing through the

More information

1. Vectors.

1. Vectors. 1. Vectors 1.1 Vectors and Matrices Linear algebra is concerned with two basic kinds of quantities: vectors and matrices. 1.1 Vectors and Matrices Scalars and Vectors - Scalar: a numerical value denoted

More information

MATH 12 CLASS 4 NOTES, SEP

MATH 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 information

Exercises for Multivariable Differential Calculus XM521

Exercises for Multivariable Differential Calculus XM521 This document lists all the exercises for XM521. The Type I (True/False) exercises will be given, and should be answered, online immediately following each lecture. The Type III exercises are to be done

More information

6. Linear Transformations.

6. Linear Transformations. 6. Linear Transformations 6.1. Matrices as Transformations A Review of Functions domain codomain range x y preimage image http://en.wikipedia.org/wiki/codomain 6.1. Matrices as Transformations A Review

More information

,, rectilinear,, spherical,, cylindrical. (6.1)

,, rectilinear,, spherical,, cylindrical. (6.1) Lecture 6 Review of Vectors Physics in more than one dimension (See Chapter 3 in Boas, but we try to take a more general approach and in a slightly different order) Recall that in the previous two lectures

More information

Introduction. Introduction (2) Easy Problems for Vectors 7/13/2011. Chapter 4. Vector Tools for Graphics

Introduction. Introduction (2) Easy Problems for Vectors 7/13/2011. Chapter 4. Vector Tools for Graphics Introduction Chapter 4. Vector Tools for Graphics In computer graphics, we work with objects defined in a three dimensional world (with 2D objects and worlds being just special cases). All objects to be

More information

MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors.

MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. MATH 304 Linear Algebra Lecture 20: The Gram-Schmidt process (continued). Eigenvalues and eigenvectors. Orthogonal sets Let V be a vector space with an inner product. Definition. Nonzero vectors v 1,v

More information

Solution. The relationship between cartesian coordinates (x, y) and polar coordinates (r, θ) is given by. (x, y) = (r cos θ, r sin θ).

Solution. The relationship between cartesian coordinates (x, y) and polar coordinates (r, θ) is given by. (x, y) = (r cos θ, r sin θ). Problem 1. Let p 1 be the point having polar coordinates r = 1 and θ = π. Let p 2 be the point having polar coordinates r = 1 and θ = π/2. Find the Euclidean distance between p 1 and p 2. The relationship

More information

MATH 1210 Assignment 3 Solutions 17R-T2

MATH 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 information

Calculus III (MAC )

Calculus III (MAC ) Calculus III (MAC2-) Test (25/9/7) Name (PRINT): Please show your work. An answer with no work receives no credit. You may use the back of a page if you need more space for a problem. You may not use any

More information

MTAEA Vectors in Euclidean Spaces

MTAEA Vectors in Euclidean Spaces School of Economics, Australian National University January 25, 2010 Vectors. Economists usually work in the vector space R n. A point in this space is called a vector, and is typically defined by its

More information

GEOMETRY AND VECTORS

GEOMETRY AND VECTORS GEOMETRY AND VECTORS Distinguishing Between Points in Space One Approach Names: ( Fred, Steve, Alice...) Problem: distance & direction must be defined point-by-point More elegant take advantage of geometry

More information

Mathematics 2203, Test 1 - Solutions

Mathematics 2203, Test 1 - Solutions Mathematics 220, Test 1 - Solutions F, 2010 Philippe B. Laval Name 1. Determine if each statement below is True or False. If it is true, explain why (cite theorem, rule, property). If it is false, explain

More information

MATH Linear Algebra

MATH Linear Algebra MATH 304 - Linear Algebra In the previous note we learned an important algorithm to produce orthogonal sequences of vectors called the Gramm-Schmidt orthogonalization process. Gramm-Schmidt orthogonalization

More information

Milford Public Schools Curriculum. Department: Mathematics Course Name: Precalculus Level 1

Milford Public Schools Curriculum. Department: Mathematics Course Name: Precalculus Level 1 Milford Public Schools Curriculum Department: Mathematics Course Name: Precalculus Level 1 UNIT 1 Unit Description: Students will construct polynomial graphs with zeros and end behavior, and apply limit

More information

Linear Algebra Homework and Study Guide

Linear Algebra Homework and Study Guide Linear Algebra Homework and Study Guide Phil R. Smith, Ph.D. February 28, 20 Homework Problem Sets Organized by Learning Outcomes Test I: Systems of Linear Equations; Matrices Lesson. Give examples of

More information

ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 3 - Thurs 5th Oct 2017 Vectors and 3D geometry

ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 3 - Thurs 5th Oct 2017 Vectors and 3D geometry ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 3 - Thurs 5th Oct 2017 Vectors and 3D geometry So far, all our calculus has been two-dimensional, involving only x and y. Nature is threedimensional,

More information

Vectors a vector is a quantity that has both a magnitude (size) and a direction

Vectors a vector is a quantity that has both a magnitude (size) and a direction Vectors In physics, a vector is a quantity that has both a magnitude (size) and a direction. Familiar examples of vectors include velocity, force, and electric field. For any applications beyond one dimension,

More information

(arrows denote positive direction)

(arrows denote positive direction) 12 Chapter 12 12.1 3-dimensional Coordinate System The 3-dimensional coordinate system we use are coordinates on R 3. The coordinate is presented as a triple of numbers: (a,b,c). In the Cartesian coordinate

More information

CONVERSION OF COORDINATES BETWEEN FRAMES

CONVERSION OF COORDINATES BETWEEN FRAMES ECS 178 Course Notes CONVERSION OF COORDINATES BETWEEN FRAMES Kenneth I. Joy Institute for Data Analysis and Visualization Department of Computer Science University of California, Davis Overview Frames

More information

MATH 1020 WORKSHEET 12.1 & 12.2 Vectors in the Plane

MATH 1020 WORKSHEET 12.1 & 12.2 Vectors in the Plane MATH 100 WORKSHEET 1.1 & 1. Vectors in the Plane Find the vector v where u =, 1 and w = 1, given the equation v = u w. Solution. v = u w =, 1 1, =, 1 +, 4 =, 1 4 = 0, 5 Find the magnitude of v = 4, 3 Solution.

More information

The geometry of least squares

The geometry of least squares The geometry of least squares We can think of a vector as a point in space, where the elements of the vector are the coordinates of the point. Consider for example, the following vector s: t = ( 4, 0),

More information

Lecture 2: Vector-Vector Operations

Lecture 2: Vector-Vector Operations Lecture 2: Vector-Vector Operations Vector-Vector Operations Addition of two vectors Geometric representation of addition and subtraction of vectors Vectors and points Dot product of two vectors Geometric

More information

Curriculum Correlation

Curriculum 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 information

Math (P)Review Part II:

Math (P)Review Part II: Math (P)Review Part II: Vector Calculus Computer Graphics Assignment 0.5 (Out today!) Same story as last homework; second part on vector calculus. Slightly fewer questions Last Time: Linear Algebra Touched

More information

Inner Product Spaces 6.1 Length and Dot Product in R n

Inner Product Spaces 6.1 Length and Dot Product in R n Inner Product Spaces 6.1 Length and Dot Product in R n Summer 2017 Goals We imitate the concept of length and angle between two vectors in R 2, R 3 to define the same in the n space R n. Main topics are:

More information

Vector Geometry. Chapter 5

Vector Geometry. Chapter 5 Chapter 5 Vector Geometry In this chapter we will look more closely at certain geometric aspects of vectors in R n. We will first develop an intuitive understanding of some basic concepts by looking at

More information

1.2 LECTURE 2. Scalar Product

1.2 LECTURE 2. Scalar Product 6 CHAPTER 1. VECTOR ALGEBRA Pythagean theem. cos 2 α 1 + cos 2 α 2 + cos 2 α 3 = 1 There is a one-to-one crespondence between the components of the vect on the one side and its magnitude and the direction

More information

NOTES ON LINEAR ALGEBRA CLASS HANDOUT

NOTES ON LINEAR ALGEBRA CLASS HANDOUT NOTES ON LINEAR ALGEBRA CLASS HANDOUT ANTHONY S. MAIDA CONTENTS 1. Introduction 2 2. Basis Vectors 2 3. Linear Transformations 2 3.1. Example: Rotation Transformation 3 4. Matrix Multiplication and Function

More information

1 Vector Geometry in Two and Three Dimensions

1 Vector Geometry in Two and Three Dimensions 1 Vector Geometry in Two and Three Dimensions 1.1 Vectors in Two Dimensions You ve probably heard of vectors as objects with both magnitude and direction, or something along these lines. Another way to

More information

4.1 Distance and Length

4.1 Distance and Length Chapter Vector Geometry In this chapter we will look more closely at certain geometric aspects of vectors in R n. We will first develop an intuitive understanding of some basic concepts by looking at vectors

More information

2. Force Systems. 2.1 Introduction. 2.2 Force

2. Force Systems. 2.1 Introduction. 2.2 Force 2. Force Systems 2.1 Introduction 2.2 Force - A force is an action of one body on another. - A force is an action which tends to cause acceleration of a body (in dynamics). - A force is a vector quantity.

More information

Linear Algebra. Min Yan

Linear Algebra. Min Yan Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................

More information

Calculus Vector Principia Mathematica. Lynne Ryan Associate Professor Mathematics Blue Ridge Community College

Calculus Vector Principia Mathematica. Lynne Ryan Associate Professor Mathematics Blue Ridge Community College Calculus Vector Principia Mathematica Lynne Ryan Associate Professor Mathematics Blue Ridge Community College Defining a vector Vectors in the plane A scalar is a quantity that can be represented by a

More information

MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.

MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation

More information

Linear Algebra II. 7 Inner product spaces. Notes 7 16th December Inner products and orthonormal bases

Linear Algebra II. 7 Inner product spaces. Notes 7 16th December Inner products and orthonormal bases MTH6140 Linear Algebra II Notes 7 16th December 2010 7 Inner product spaces Ordinary Euclidean space is a 3-dimensional vector space over R, but it is more than that: the extra geometric structure (lengths,

More information

Definition 1. A set V is a vector space over the scalar field F {R, C} iff. there are two operations defined on V, called vector addition

Definition 1. A set V is a vector space over the scalar field F {R, C} iff. there are two operations defined on V, called vector addition 6 Vector Spaces with Inned Product Basis and Dimension Section Objective(s): Vector Spaces and Subspaces Linear (In)dependence Basis and Dimension Inner Product 6 Vector Spaces and Subspaces Definition

More information

1 Differentiable manifolds and smooth maps. (Solutions)

1 Differentiable manifolds and smooth maps. (Solutions) 1 Differentiable manifolds and smooth maps Solutions Last updated: March 17 2011 Problem 1 The state of the planar pendulum is entirely defined by the position of its moving end in the plane R 2 Since

More information

PreCalculus Honors Curriculum Pacing Guide First Half of Semester

PreCalculus Honors Curriculum Pacing Guide First Half of Semester Unit 1 Introduction to Trigonometry (9 days) First Half of PC.FT.1 PC.FT.2 PC.FT.2a PC.FT.2b PC.FT.3 PC.FT.4 PC.FT.8 PC.GCI.5 Understand that the radian measure of an angle is the length of the arc on

More information

SUPPLEMENT I. Example. Graph the vector 4, 3. Definition. Given two points A(x 1, y 1 ) and B(x 2, y 2 ), the vector represented by # AB is # AB =,

SUPPLEMENT I. Example. Graph the vector 4, 3. Definition. Given two points A(x 1, y 1 ) and B(x 2, y 2 ), the vector represented by # AB is # AB =, SUPPLEMENT I 1. Vectors Definition. A vector is a quantity that has both a magnitude and a direction. A twodimensional vector is an ordered pair a = a 1, a 2 of real numbers. The numbers a 1 and a 2 are

More information

and Calculus and Vectors

and 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 information

Page 52. Lecture 3: Inner Product Spaces Dual Spaces, Dirac Notation, and Adjoints Date Revised: 2008/10/03 Date Given: 2008/10/03

Page 52. Lecture 3: Inner Product Spaces Dual Spaces, Dirac Notation, and Adjoints Date Revised: 2008/10/03 Date Given: 2008/10/03 Page 5 Lecture : Inner Product Spaces Dual Spaces, Dirac Notation, and Adjoints Date Revised: 008/10/0 Date Given: 008/10/0 Inner Product Spaces: Definitions Section. Mathematical Preliminaries: Inner

More information

MECH 576 Geometry in Mechanics November 30, 2009 Kinematics of Clavel s Delta Robot

MECH 576 Geometry in Mechanics November 30, 2009 Kinematics of Clavel s Delta Robot MECH 576 Geometry in Mechanics November 3, 29 Kinematics of Clavel s Delta Robot The DELTA Robot DELTA, a three dimensional translational manipulator, appears below in Fig.. Figure : Symmetrical (Conventional)

More information