Math 241, Exam 1 Information.


 Randolph Griffin
 2 years ago
 Views:
Transcription
1 Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections , The corresponding assigned homework problems (see boylan/sccourses/241fa12/241.html) At minimum, you need to understand how to do the homework problems. Topic List (not necessarily comprehensive): You will need to know: theorems, results, and definitions from class. 12.1: 3dimensional space. Distance formula: d(p 1, P 2 ) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 + (z 2 z 1 ) 2. Sphere with center (x 0, y 0, z 0 ) and radius r: 12.2: Vectors. {(x, y, z) : (x x 0 ) 2 + (y y 0 ) 2 + (z z 0 ) 2 = r 2 }. Vectors have length (magnitude) and direction. Operations: 1. Addition. (a) Algebraically: u 1, u 2, u 3 + v 1, v 2, v 3 = u 1 + v 1, u 2 + v 2, u 3 + v 3. (b) Geometrically: Parallelogram Law. (c) Properties: closure, zero element, associativity, inverses ( u = ( 1) u), commutativity. 2. Scalar multiplication. (a) Algebraically: k u 1, u 2, u 3 = ku 1, ku 2, ku 3. (b) Geometrically: k u scales u by a factor of k ; if k < 0, then k u points in the direction opposite u. (c) Properties: closure, associativity, distributivity.
2 Length. v 1, v 2, v 3 = v1 2 + v2 2 + v3 2 Notes. 1. For all v R 3, we have v 0 and v = 0 v = For k R, we have k v = k v. Unit vectors. A vector v is a unit vector if and only if v = Standard basis unit vectors: i = 1, 0, 0, j = 0, 1, 0, k = 0, 0, The normalization of v R 3 is the unit vector v v. Parallel vectors. Two nonzero vectors u and v are parallel ( u v) if and only if there exists k 0 in R with k u = v. 12.3: The dot product. Definition: u 1, u 2, u 3 v 1, v 2, v 3 = u 1 v 1 + u 2 v 2 + u 3 v 3. The input consists of two vectors; the output is a scalar. Properties. 1. Commutative, distributive. 2. u u = u Let 0 θ π be the angle between vectors u and v. Then we have cos θ = u v u v. Orthogonality. Two vectors u and v are orthogonal if and only if u v = 0; geometrically, this holds if and only if the vectors are perpendicular: the angle between them is π/2. ( 1. The orthogonal projection of v on v b is proj b v = b ) b. b 2 2. The component of v on b is comp b v = proj b v = v b b.
3 12.4: The cross product. Definition: Let u = u 1, u 2, u 3, v = v 1, v 2, v 3. Then we have i j k u v = u 1 u 2 u 3 v 1 v 2 v 3 = (u 2v 3 v 2 u 3 ) i (u 1 v 3 v 1 u 3 ) j + (u 1 v 2 v 1 u 2 ) k Two vectors are input; the output is a vector. Facts. 1. Distributive. 2. Not associative in general. 3. u u = Anticommutative: u v = v u. 5. i j = k, j k = i, k i = j. The other products of these vectors are obtainable from anticommutativity. Geometry. 1. u v is orthogonal to both u and v. Of two possible directions, the correct one is obtained via the righthand rule. 2. Let 0 θ π be the angle between u and v. Then we have sin θ = 3. The area of the parallelogram determined by u and v is u v. 4. The vectors u and v are parallel ( u v) if and only if u v = 0. u v u v. Triple product: Let u = u 1, u 2, u 3, v = v 1, v 2, v 3, w = w 1, w 2, w 3. Then we have u 1 u 2 u 3 u ( v w) = v 1 v 2 v 3 w 1 w 2 w 3. Moreover: 1. This value gives the volume of the parallelepiped determined by u, v, and w. 2. This value is zero if and only if the three vectors are coplanar.
4 12.5: Equations of lines and planes. I. Lines. A line L is determined by a direction vector v = a, b, c and a point P 0 = (x 0, y 0, z 0 ) on the line. The equation of a line can take several forms (a) Parametric form. x = x 0 + at, y = y 0 + bt, z = z 0 + ct, t R. (b) Vector form. r = r 0 + t v. Here, we have r = x, y, z and r 0 = x 0, y 0, z 0. (c) Symmetric form. Suppose that none of a, b, c is zero. Then we have newline x x 0 = y y 0 = z z 0. Suitable adjustments are required when one or more a b c of the components of v is zero. Skew lines. Two lines in R 3 are skew if and only if they are nonparallel lines lying in parallel planes. Facts. Let L 1 and L 2 be lines with direction vectors v 1 and v 2. (a) The lines L 1 and L 2 are parallel if and only if their direction vectors are; this happens if and only if there exists k R with k v 1 = v 2. (b) The lines L 1 and L 2 are perpendicular if and only if their direction vectors are orthogonal: u v = 0. (c) Two lines are skew if they are nonintersecting and nonparallel. To determine whether two nonparallel lines intersect requires one to solve a system of three equations in two variables. If the system has a solution, the lines intersect at the point given by the solution; if the system has no solution, then the lines are skew. II. Planes. A plane Q is determined by a normal vector n = a, b, c which measures the tilt of Q in R 3 and a point P 0 = (x 0, y 0, z 0 ) on Q. The normal vector is perpendicular to Q. The equation of the plane Q is (a) Q : a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0. (b) Q : ax + by + cz = d, where d = ax 0 + by 0 + cz 0. Observations. (a) Planes Q 1 and Q 2 are parallel if and only if their normal vectors are: n 1 n 2. (b) Planes Q 1 and Q 2 are perpendicular if and only if their normal vectors are: n 1 n 2. (c) Line L is parallel to plane Q if and only if the direction vector of L is perpendicular to the normal of Q: n v. (d) Line L is perpendicular to plane Q if and only if the direction vector of L is parallel to the normal of Q: n v.
5 Distance. Let P 0 = (x 0, y 0, z 0 ) and let Q : ax + by + cz + d = 0. Then the distance from P 0 to Q is d(p 0, Q) = ax 0 + by 0 + cz 0 + d. a2 + b 2 + c 2 One can use this formula to find the distance between the following objects (assumed to be nonintersecting so the distance makes sense): a point and a plane; a point and a line; a line and a plane; two planes; 14.1: Functions of several variables. Given a function z = f(x, y), be able to sketch or describe the domain, the graph, and the level curves. 14.2: Limits and continuity. Limits along curves: Let C be a smooth curve in R 2 through the point P 0 = (x 0, y 0 ), and suppose that C has parametric equations C : x = x(t), y = y(t), t R, with x 0 = x(t 0 ), y 0 = y(t 0 ) for some t 0. Then we have lim f(x, y) = lim f(x(t), y(t)) (x,y) (x 0,y 0 ) along C t t0 Remarks: We have lim f(x, y) exists if and only if lim f(x, y) exists and gives a common (x,y) (x 0,y 0 ) (x,y) (x 0,y 0 ) along C value for all curves C through (x 0, y 0 ). Suppose that there exist curves C 1 and C 2 through (x 0, y 0 ) with Then lim f(x, y) does not exist. (x,y) (x 0,y 0 ) lim f(x, y) lim f(x, y) (x,y) (x 0,y 0 ) along C 1 (x,y) (x 0,y 0 ) along C 2
6 Definition: The function f(x, y) is continuous at (x 0, y 0 ) if and only if 1. (x 0, y 0 ) is in the domain of f and 2. lim f(x, y) = f(x 0, y 0 ). (x,y) (x 0,y 0 ) 14.3: Partial derivatives. The partial derivative of z = f(x, y) with respect to x is obtained by differentiating with respect to x while holding y constant; similarly, the partial with respect to y is obtained by differentiating with respect to y while holding x constant. Notation: Let z = f(x, y). We write f x (x, y) = f x (x, y) = x z, f y(x, y) = f y (x, y) = y z. Fact: Let z = f(x, y), and let P 0 = (x 0, y 0 ). Then f x (x 0, y 0 ) gives the slope of the line tangent to the surface at P 0 in the direction of the unit vector i; similarly, f y (x 0, y 0 ) gives the slope of the line tangent to the surface at P 0 in the direction of j. Higher order partials: 2 f x = 2 x ( ) f = f xx, x 2 f y 2 = y ( ) f = f yy. y Clairaut s Theorem implies that 2 f y x = f xy = f yx = 2 f x y
MAT 1341: REVIEW II SANGHOON BAEK
MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and
More informationEquations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More informationGeometric description of the cross product of the vectors u and v. The cross product of two vectors is a vector! u x v is perpendicular to u and v
12.4 Cross Product Geometric description of the cross product of the vectors u and v The cross product of two vectors is a vector! u x v is perpendicular to u and v The length of u x v is uv u v sin The
More information12.5 Equations of Lines and Planes
Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P
More informationL 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More information13.4 THE CROSS PRODUCT
710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product
More informationSection 9.5: Equations of Lines and Planes
Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 35 odd, 237 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
More informationLINES AND PLANES CHRIS JOHNSON
LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3space, as well as define the angle between two nonparallel planes, and determine the distance
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More informationProblem set on Cross Product
1 Calculate the vector product of a and b given that a= 2i + j + k and b = i j k (Ans 3 j  3 k ) 2 Calculate the vector product of i  j and i + j (Ans ) 3 Find the unit vectors that are perpendicular
More information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationSection 2.4: Equations of Lines and Planes
Section.4: Equations of Lines and Planes An equation of three variable F (x, y, z) 0 is called an equation of a surface S if For instance, (x 1, y 1, z 1 ) S if and only if F (x 1, y 1, z 1 ) 0. x + y
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two nonzero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More informationv 1 v 3 u v = (( 1)4 (3)2, [1(4) ( 2)2], 1(3) ( 2)( 1)) = ( 10, 8, 1) (d) u (v w) = (u w)v (u v)w (Relationship between dot and cross product)
0.1 Cross Product The dot product of two vectors is a scalar, a number in R. Next we will define the cross product of two vectors in 3space. This time the outcome will be a vector in 3space. Definition
More informationSection 13.5 Equations of Lines and Planes
Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines  specifically, tangent lines.
More informationVectors, Gradient, Divergence and Curl.
Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use
More informationExam 1 Sample Question SOLUTIONS. y = 2x
Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can
More information28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z
28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition
More informationName Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155
Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate
More informationv w is orthogonal to both v and w. the three vectors v, w and v w form a righthanded set of vectors.
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 informationSection 11.4: Equations of Lines and Planes
Section 11.4: Equations of Lines and Planes Definition: The line containing the point ( 0, 0, 0 ) and parallel to the vector v = A, B, C has parametric equations = 0 + At, = 0 + Bt, = 0 + Ct, where t R
More informationCross product and determinants (Sect. 12.4) Two main ways to introduce the cross product
Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Geometrical definition Properties Expression in components. Definition in components Properties Geometrical expression.
More informationChapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis
Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationSection 12.6: Directional Derivatives and the Gradient Vector
Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate
More information1.5 Equations of Lines and Planes in 3D
40 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3D Recall that given a point P = (a, b, c), one can draw a vector from
More informationDot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product
Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationSection 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables
The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,
More informationCHAPTER FIVE. 5. Equations of Lines in R 3
118 CHAPTER FIVE 5. Equations of Lines in R 3 In this chapter it is going to be very important to distinguish clearly between points and vectors. Frequently in the past the distinction has only been a
More informationHOMEWORK 4 SOLUTIONS. All questions are from Vector Calculus, by Marsden and Tromba
HOMEWORK SOLUTIONS All questions are from Vector Calculus, by Marsden and Tromba Question :..6 Let w = f(x, y) be a function of two variables, and let x = u + v, y = u v. Show that Solution. By the chain
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level  MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: 97 Mathematics
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals,
More informationDetermine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s
Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,
More informationTwo vectors are equal if they have the same length and direction. They do not
Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More informationLecture L3  Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3  Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More informationReview Sheet for Test 1
Review Sheet for Test 1 Math 26100 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And
More information10.5. Click here for answers. Click here for solutions. EQUATIONS OF LINES AND PLANES. 3x 4y 6z 9 4, 2, 5. x y z. z 2. x 2. y 1.
SECTION EQUATIONS OF LINES AND PLANES 1 EQUATIONS OF LINES AND PLANES A Click here for answers. S Click here for solutions. 1 Find a vector equation and parametric equations for the line passing through
More informationSolutions to Homework 5
Solutions to Homework 5 1. Let z = f(x, y) be a twice continously differentiable function of x and y. Let x = r cos θ and y = r sin θ be the equations which transform polar coordinates into rectangular
More information(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,
Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We
More informationi=(1,0), j=(0,1) in R 2 i=(1,0,0), j=(0,1,0), k=(0,0,1) in R 3 e 1 =(1,0,..,0), e 2 =(0,1,,0),,e n =(0,0,,1) in R n.
Length, norm, magnitude of a vector v=(v 1,,v n ) is v = (v 12 +v 22 + +v n2 ) 1/2. Examples v=(1,1,,1) v =n 1/2. Unit vectors u=v/ v corresponds to directions. Standard unit vectors i=(1,0), j=(0,1) in
More information1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.
.(6pts Find symmetric equations of the line L passing through the point (, 5, and perpendicular to the plane x + 3y z = 9. (a x = y + 5 3 = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3
More informationThe Dot and Cross Products
The Dot and Cross Products Two common operations involving vectors are the dot product and the cross product. Let two vectors =,, and =,, be given. The Dot Product The dot product of and is written and
More informationPractice Final Math 122 Spring 12 Instructor: Jeff Lang
Practice Final Math Spring Instructor: Jeff Lang. Find the limit of the sequence a n = ln (n 5) ln (3n + 8). A) ln ( ) 3 B) ln C) ln ( ) 3 D) does not exist. Find the limit of the sequence a n = (ln n)6
More informationSOLUTIONS. f x = 6x 2 6xy 24x, f y = 3x 2 6y. To find the critical points, we solve
SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives
More information2.1 Three Dimensional Curves and Surfaces
. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two or threedimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The
More informationJim Lambers MAT 169 Fall Semester 200910 Lecture 25 Notes
Jim Lambers MAT 169 Fall Semester 00910 Lecture 5 Notes These notes correspond to Section 10.5 in the text. Equations of Lines A line can be viewed, conceptually, as the set of all points in space that
More informationMA261A Calculus III 2006 Fall Homework 3 Solutions Due 9/22/2006 8:00AM
MA6A Calculus III 6 Fall Homework Solutions Due 9//6 :AM 9. # Find the parametric euation and smmetric euation for the line of intersection of the planes + + z = and + z =. To write down a line euation,
More informationVector Algebra CHAPTER 13. Ü13.1. Basic Concepts
CHAPTER 13 ector Algebra Ü13.1. Basic Concepts A vector in the plane or in space is an arrow: it is determined by its length, denoted and its direction. Two arrows represent the same vector if they have
More informationLINES AND PLANES IN R 3
LINES AND PLANES IN R 3 In this handout we will summarize the properties of the dot product and cross product and use them to present arious descriptions of lines and planes in three dimensional space.
More informationProblem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.
Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve
More informationThe Vector or Cross Product
The Vector or ross Product 1 ppendix The Vector or ross Product We saw in ppendix that the dot product of two vectors is a scalar quantity that is a maximum when the two vectors are parallel and is zero
More informationPractice Problems for Midterm 2
Practice Problems for Midterm () For each of the following, find and sketch the domain, find the range (unless otherwise indicated), and evaluate the function at the given point P : (a) f(x, y) = + 4 y,
More informationMATH 275: Calculus III. Lecture Notes by Angel V. Kumchev
MATH 275: Calculus III Lecture Notes by Angel V. Kumchev Contents Preface.............................................. iii Lecture 1. ThreeDimensional Coordinate Systems..................... 1 Lecture
More informationVector Math Computer Graphics Scott D. Anderson
Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about
More informationMathematics Notes for Class 12 chapter 10. Vector Algebra
1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is nonnegative
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More informationGeometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi
Geometry of Vectors Carlo Tomasi This note explores the geometric meaning of norm, inner product, orthogonality, and projection for vectors. For vectors in threedimensional space, we also examine the
More informationLinear Algebra Test 2 Review by JC McNamara
Linear Algebra Test 2 Review by JC McNamara 2.3 Properties of determinants: det(a T ) = det(a) det(ka) = k n det(a) det(a + B) det(a) + det(b) (In some cases this is true but not always) A is invertible
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationMATH 304 Linear Algebra Lecture 24: Scalar product.
MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B A A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent
More informationJUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson
JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3
More informationReview of Vector Analysis in Cartesian Coordinates
R. evicky, CBE 6333 Review of Vector Analysis in Cartesian Coordinates Scalar: A quantity that has magnitude, but no direction. Examples are mass, temperature, pressure, time, distance, and real numbers.
More information= y y 0. = z z 0. (a) Find a parametric vector equation for L. (b) Find parametric (scalar) equations for L.
Math 21a Lines and lanes Spring, 2009 Lines in Space How can we express the equation(s) of a line through a point (x 0 ; y 0 ; z 0 ) and parallel to the vector u ha; b; ci? Many ways: as parametric (scalar)
More informationRecall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula:
Chapter 7 Div, grad, and curl 7.1 The operator and the gradient: Recall that the gradient of a differentiable scalar field ϕ on an open set D in R n is given by the formula: ( ϕ ϕ =, ϕ,..., ϕ. (7.1 x 1
More informationMath 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t
Math 241 Lines and Planes (Solutions) The equations for planes P 1, P 2 and P are P 1 : x 2y + z = 7 P 2 : x 4y + 5z = 6 P : (x 5) 2(y 6) + (z 7) = 0 The equations for lines L 1, L 2, L, L 4 and L 5 are
More informationSurface Normals and Tangent Planes
Surface Normals and Tangent Planes Normal and Tangent Planes to Level Surfaces Because the equation of a plane requires a point and a normal vector to the plane, nding the equation of a tangent plane to
More informationApr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa
Calculus with Algebra and Trigonometry II Lecture 23 Final Review: Curve sketching and parametric equations Apr 23, 2015 Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23,
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationa a. θ = cos 1 a b ) b For nonzero vectors a and b, then the component of b along a is given as comp
Textbook Assignment 4 Your Name: LAST NAME, FIRST NAME (YOUR STUDENT ID: XXXX) Your Instructors Name: Prof. FIRST NAME LAST NAME YOUR SECTION: MATH 0300 XX Due Date: NAME OF DAY, MONTH DAY, YEAR. SECTION
More informationdiscuss how to describe points, lines and planes in 3 space.
Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More informationNotice that v v w (4)( 15) ( 3)( 20) (0)(2) ( 2)( 15) (2)( 20) (5)(2)
The Cross Product When discussing the dot product, we showed how two vectors can be combined to get a number. Here, we shall see another way of combining vectors, this time resulting in a vector. This
More information... ... . (2,4,5).. ...
12 Three Dimensions ½¾º½ Ì ÓÓÖ Ò Ø ËÝ Ø Ñ So far wehave been investigatingfunctions ofthe form y = f(x), withone independent and one dependent variable Such functions can be represented in two dimensions,
More informationOrthogonal Projections and Orthonormal Bases
CS 3, HANDOUT A, 3 November 04 (adjusted on 7 November 04) Orthogonal Projections and Orthonormal Bases (continuation of Handout 07 of 6 September 04) Definition (Orthogonality, length, unit vectors).
More informationSome Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)
Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving
More informationx(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3
CORE 4 Summary Notes Rational Expressions Factorise all expressions where possible Cancel any factors common to the numerator and denominator x + 5x x(x + 5) x 5 (x + 5)(x 5) x x 5 To add or subtract 
More informationCalculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum
Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic
More information1. Vectors and Matrices
E. 8.02 Exercises. Vectors and Matrices A. Vectors Definition. A direction is just a unit vector. The direction of A is defined by dir A = A, (A 0); A it is the unit vector lying along A and pointed like
More informationa.) Write the line 2x  4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a
Bellwork a.) Write the line 2x  4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a c.) Find the slope of the line perpendicular to part b or a May 8 7:30 AM 1 Day 1 I.
More informationLimits and Continuity
Math 20C Multivariable Calculus Lecture Limits and Continuity Slide Review of Limit. Side limits and squeeze theorem. Continuous functions of 2,3 variables. Review: Limits Slide 2 Definition Given a function
More informationDefinition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.
6.1 Vectors in the Plane PreCalculus 6.1 VECTORS IN THE PLANE Learning Targets: 1. Find the component form and the magnitude of a vector.. Perform addition and scalar multiplication of two vectors. 3.
More informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More informationMATHEMATICS (CLASSES XI XII)
MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationAverage rate of change of y = f(x) with respect to x as x changes from a to a + h:
L151 Lecture 15: Section 3.4 Definition of the Derivative Recall the following from Lecture 14: For function y = f(x), the average rate of change of y with respect to x as x changes from a to b (on [a,
More informationChapter 1: Essentials of Geometry
Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,
More informationExample SECTION 131. XAXIS  the horizontal number line. YAXIS  the vertical number line ORIGIN  the point where the xaxis and yaxis cross
CHAPTER 13 SECTION 131 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants XAXIS  the horizontal
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More information2.1 Functions. 2.1 J.A.Beachy 1. from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair
2.1 J.A.Beachy 1 2.1 Functions from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 21. The Vertical Line Test from calculus says that a curve in the xyplane
More information6. Vectors. 1 20092016 Scott Surgent (surgent@asu.edu)
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
More information