- parametric equations for the line, z z 0 td 3 or if d 1 0, d 2 0andd 3 0, - symmetric equations of the line.
|
|
- Brett Williamson
- 5 years ago
- Views:
Transcription
1 Lines and Planes in Space -(105) Questions: What do we need to know to determine a line in space? What are the fms of a line? If two lines are not parallel in space, must they be intersect as two lines in plane? What do we need to know to determine a plane in space? 1 Lines in Space Consider a line which passes through the point P 0 x 0, y 0, z 0 in the direction d d 1, d 2, d LetP x,y,z be a point on the line Then the vect P 0 P is parallel to the direction vect d,thatis,p 0 P t d where t is a scalar Hence, P 0 P x x 0, y y 0, z z 0 t d td 1, td 2, td x x 0 td 1 y y 0 td 2 - parametric equations f the line, z z 0 td if d 1 0, d 2 0andd 0, x x 0 y y 0 - symmetric equations of the line d 1 d 2 d In the case where d 1 0, the line is in a plane which is parallel to the yz-plane so the line equation is x x 0 and y y 0 Similarly, when d 2 0, the line equation is y y 0 and d 2 d x x 0,andwhend d 1 d 0, the line equation is z z 0 and x x 0 y y 0 d 1 d 2 Definition: Let two lines L 1 and L 2 be in the direction of d 1 and d 2 Then L 1 and L 2 are parallel if d 1 c d 2 If L 1 and L 2 intersect, thentheangle between L 1 and L 2 is the angle between d 1 and d 2 If L 1 and L 2 are thogonal, then d 1 d 2 0 If L 1 and L 2 are not parallel and not intersecting, then we say they are skew Note that two lines intersect if and only if both lines have a common point Example a Find an equation of the line passing through the point P 1,5,2 and parallel to the vect d,,7 b Determine also where the line intersects the yz plane c Find the line passing through the point P and parallel to the line: x 2 1 y 1 z d Find the line passing through the point P and perpendicular to lines: x 2 1 y 1 z x 1 1t and y 2 2t z 1 t 1
2 a The parametric equations f this line: x 1 t y 5 t z 2 7t b The line intersects with the yz plane when x 0 Then 1 t, t 1,and y 5 1, z Hence the line intersects the yz plane at the point 0, 17, 1 x 1 2t c The direction vect d 2,,1 So the parametric equations of the line: y 5 t z 2 t i j k d The direction vect d 2,,1 1,2, 2 1 7i 5 j 7 k 7,5, The parametric equations of the line is: x 1 7t y 5 5t z 2 7t Example Find the equation of the line through points 1,2, 1 and 5,, d 5 1, 2, 1, 5,5 The parametric equations of the line is x 1 t y 2 5t z 1 5t Example Determine if the lines x 2 t y 1 2t z 5 2t, and x 1 s y 2 s z 1 s are parallel, skew intersect a Check if two lines are parallel: d 1 1,2,2, d 2 1, 1,, sinced 1 cd 2 two lines are not parallel b Check if two lines are intersect (have a common point): Canwefindasand a t such that x,y,z are the same f both parametric equations? Set x :1 s 2 t y :2 s 1 2t z :1 s 5 2t From equation f x we have s 1 t Now substitute s 1 t into equation f y : 1 s 2t we have 1 1 t t 2t So, two lines do not intersect Therefe, two lines are skew 2
3 2 Planes in R Simple planes: y - a plane parallel to xz-plane and passing the point 0,,0 x 2 - a plane parallel to yz-plane and passing the point 2,0,0 z - a plane parallel to xy-plane and passing the point 0,0, ; y x 2 z Let P 0 x 0, y 0, z 0 be a point and n n 1,n 2,n beavectinr Find the equation of the plane that contains P 0 and the vect n is nmal to (n is thogonal to the plane) Let P x, y, z be a point in the plane Then we know the vect P 0 P and the nmal vect n are thogonal, that is, n P 0 P n 1, n 2, n x x 0, y y 0, z z 0 0 n 1 x x 0 n 2 y y 0 n z z the equation f the plane n 1 x n 2 y n z n 1 x 0 n 2 y 0 n z 0 n 1 x n 2 y n z n 1 x 0 n 2 y 0 n z 0 0 Clearly, this equation is linear in R Let m 1 x x 0 m 2 y y 0 m z z 0 0andn 1 x x 0 n 2 y y 0 n z z 0 0betwo planes Let m m 1,m 2, m and n n 1,n 2, n Note that: Two planes are parallel if and only if two nmal vects m and n are parallel, that is: m c n f a constant c Two planes are thogonal if and only if two nmal vects m and n are thogonal, that is, m n 0 Example Find an equation of the plane containing the point 1, 2, with nmal vect,5,6 The equation of the plane: x 1 5 y 2 6 z 0 Example Find the plane containing the three points P 1,2,2, Q 2, 1,, and R,5, 2
4 i j k n PQ PR 1,,2 2,, 1 2 6i 8j 9k 6,8,9 2 The equation of the plane: 6 x 1 8 y 2 9 z 2 0 Example Sketch the place x 2y z 1 x 2y z 1 in the 1st octant x 2y z 1 Example Find an equation f the plane passing through the point 1,, 5 and parallel to the plane defined by 2x 5y 7z 12 The nmal vect of the plane is n 2, 5,7 Then the equation of the plane: 2 x 1 5 y 7 z 5 0, 2x 5y 7z Example Find an equation of the line which intersects planes: x 2y z, and x y z 5 Solve x 2y z (1) x y z 5 (2) f xand z in terms of y : (1)-(2):6y 2z 2, y z 1, z y 1 (1): x 2y z 2y y 1 2 5y Since y is free, let y t The solution is equation x 2 5t y t z 1 t, the parametric equations f the line Example Find an equation of the plane containing the point 1,, 1 and perpendicular to the planes x y 2z 1 and 2x y z 2
5 The nmal vects of the given planes are: n 1 1,1, 2 and n 2 2,1,1 The nmal vect n of the plane is thogonal to both given planes, that is, n n 1 n 2 n i j k i 5j k The equation of the plane is: x 1 5 y z 1 0 Example Find the distance between the parallel planes: 2x y z 6, and x 6y 2z 8 a Pick one point from each plane: P 1 1, 1, 7 from the 1st plane and P 2 0, 0, from the 2nd plane b Find the vect P 1 P 2 : P 1 P 2 1, 1, c Compute comp n P 1 P 2 : n 2,,1 comp n P 1 P 2 P 1P 2 n n
- parametric equations for the line, z z 0 td 3 or if d 1 0, d 2 0andd 3 0, - symmetric equations of the line.
Lines and Planes in Space -(105) Questions: 1 What is the equation of a line if we know (1) two points P x 1,y 1,z 1 and Q x 2,y 2,z 2 on the line; (2) a point P x 1,y 1,z 1 on the line and the line is
More information8. 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 informationVECTORS IN A STRAIGHT LINE
A. The Equation of a Straight Line VECTORS P3 VECTORS IN A STRAIGHT LINE A particular line is uniquely located in space if : I. It has a known direction, d, and passed through a known fixed point, or II.
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 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 informationCALCULUS 3 February 6, st TEST
MATH 400 (CALCULUS 3) Spring 008 1st TEST 1 CALCULUS 3 February, 008 1st TEST YOUR NAME: 001 A. Spina...(9am) 00 E. Wittenbn... (10am) 003 T. Dent...(11am) 004 J. Wiscons... (1pm) 005 A. Spina...(1pm)
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 information1.2 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 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 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 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 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 informationDepartment 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 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 informationCalculus 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 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 informationMath 3c Solutions: Exam 1 Fall Graph by eliiminating the parameter; be sure to write the equation you get when you eliminate the parameter.
Math c Solutions: Exam 1 Fall 16 1. Graph by eliiminating the parameter; be sure to write the equation you get when you eliminate the parameter. x tan t x tan t y sec t y sec t t π 4 To eliminate the parameter,
More informationMathematics 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 informationPractice problems for Exam 1. a b = (2) 2 + (4) 2 + ( 3) 2 = 29
Practice problems for Exam.. Given a = and b =. Find the area of the parallelogram with adjacent sides a and b. A = a b a ı j k b = = ı j + k = ı + 4 j 3 k Thus, A = 9. a b = () + (4) + ( 3)
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 informationDEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS. MATH 233 SOME SOLUTIONS TO EXAM 1 Fall 2018
DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF MASSACHUSETTS MATH SOME SOLUTIONS TO EXAM 1 Fall 018 Version A refers to the regular exam and Version B to the make-up 1. Version A. Find the center
More informationSOLUTIONS TO HOMEWORK ASSIGNMENT #2, Math 253
SOLUTIONS TO HOMEWORK ASSIGNMENT #, Math 5. Find the equation of a sphere if one of its diameters has end points (, 0, 5) and (5, 4, 7). The length of the diameter is (5 ) + ( 4 0) + (7 5) = =, so the
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 informationExam 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 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 informationDistance. Warm Ups. Learning Objectives I can find the distance between two points. Football Problem: Bailey. Watson
Distance Warm Ups Learning Objectives I can find the distance between two points. Football Problem: Bailey Watson. Find the distance between the points (, ) and (4, 5). + 4 = c 9 + 6 = c 5 = c 5 = c. Using
More informationMATH 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 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 informationvand v 3. Find the area of a parallelogram that has the given vectors as adjacent sides.
Name: Date: 1. Given the vectors u and v, find u vand v v. u= 8,6,2, v = 6, 3, 4 u v v v 2. Given the vectors u nd v, find the cross product and determine whether it is orthogonal to both u and v. u= 1,8,
More informationMATH H53 : Mid-Term-1
MATH H53 : Mid-Term-1 22nd September, 215 Name: You have 8 minutes to answer the questions. Use of calculators or study materials including textbooks, notes etc. is not permitted. Answer the questions
More information3. Interpret the graph of x = 1 in the contexts of (a) a number line (b) 2-space (c) 3-space
MA2: Prepared by Dr. Archara Pacheenburawana Exercise Chapter 3 Exercise 3.. A cube of side 4 has its geometric center at the origin and its faces parallel to the coordinate planes. Sketch the cube and
More information(a) The points (3, 1, 2) and ( 1, 3, 4) are the endpoints of a diameter of a sphere.
MATH 4 FINAL EXAM REVIEW QUESTIONS Problem. a) The points,, ) and,, 4) are the endpoints of a diameter of a sphere. i) Determine the center and radius of the sphere. ii) Find an equation for the sphere.
More informationOverview. Distances in R 3. Distance from a point to a plane. Question
Overview Yesterda we introduced equations to describe lines and planes in R 3 : r + tv The vector equation for a line describes arbitrar points r in terms of a specific point and the direction vector v.
More informationMATHEMATICS AS/M/P1 AS PAPER 1
Surname Other Names Candidate Signature Centre Number Candidate Number Examiner Comments Total Marks MATHEMATICS AS PAPER 1 Bronze Set B (Edexcel Version) CM Time allowed: 2 hours Instructions to candidates:
More informationMAT 1339-S14 Class 10 & 11
MAT 1339-S14 Class 10 & 11 August 7 & 11, 2014 Contents 8 Lines and Planes 1 8.1 Equations of Lines in Two-Space and Three-Space............ 1 8.2 Equations of Planes........................... 5 8.3 Properties
More informationReview Sheet for the Final
Review Sheet for the Final Math 6-4 4 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 the absence
More information. Let us consider the point P with coordinates y = R, z =0,0 x L. Evaluate the principal stresses and the principal stress directions.
14 3. The linear 3-D elasticity mathematical model and substituting in (3.94) gives (n 1 ) =1 (n ) ( ) t n =λ 1 ( 1 (n ) ( ) ) + λ (n ) + λ 3 ( ) t n =λ 1 +(λ λ 1 )(n ) +(λ 3 λ 1 )( ). (3.95) Since λ 1
More informationCreated by T. Madas LINE INTEGRALS. Created by T. Madas
LINE INTEGRALS LINE INTEGRALS IN 2 DIMENSIONAL CARTESIAN COORDINATES Question 1 Evaluate the integral ( x + 2y) dx, C where C is the path along the curve with equation y 2 = x + 1, from ( ) 0,1 to ( )
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 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 informationreview To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = -17
1. Revision Recall basic terms of algebraic expressions like Variable, Constant, Term, Coefficient, Polynomial etc. The coefficients of the terms in 4x 2 5xy + 6y 2 are Coefficient of 4x 2 is 4 Coefficient
More information1 Geometry of R Conic Sections Parametric Equations More Parametric Equations Polar Coordinates...
Contents 1 Geometry of R 2 2 1.1 Conic Sections............................................ 2 1.2 Parametric Equations........................................ 3 1.3 More Parametric Equations.....................................
More information(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 informationWorksheet A VECTORS 1 G H I D E F A B C
Worksheet A G H I D E F A B C The diagram shows three sets of equally-spaced parallel lines. Given that AC = p that AD = q, express the following vectors in terms of p q. a CA b AG c AB d DF e HE f AF
More information7.2. Exercises on lines in space
.. Exercises on lines in space Exercise : Change of support and direction vectors. Check whether P or Q are on g.. Use the point found in. to find both a new support vector and a new direction vector for
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 information(Chapter 10) (Practical Geometry) (Class VII) Question 1: Exercise 10.1 Draw a line, say AB, take a point C outside it. Through C, draw a line parallel to AB using ruler and compasses only. Answer 1: To
More information5. Introduction to Euclid s Geometry
5. Introduction to Euclid s Geometry Multiple Choice Questions CBSE TREND SETTER PAPER _ 0 EXERCISE 5.. If the point P lies in between M and N and C is mid-point of MP, then : (A) MC + PN = MN (B) MP +
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Spring 2018, WEEK 1 JoungDong Kim Week 1 Vectors, The Dot Product, Vector Functions and Parametric Curves. Section 1.1 Vectors Definition. A Vector is a quantity that
More informationExam. There are 6 problems. Your 5 best answers count. Please pay attention to the presentation of your work! Best 5
Department of Mathematical Sciences Instructor: Daiva Pucinskaite Calculus III June, 06 Name: Exam There are 6 problems. Your 5 best answers count. Please pay attention to the presentation of your work!
More informationREVIEW 2, MATH 3020 AND MATH 3030
REVIEW, MATH 300 AND MATH 3030 1. Let P = (0, 1, ), Q = (1,1,0), R(0,1, 1), S = (1,, 4). (a) Find u = PQ and v = PR. (b) Find the angle between u and v. (c) Find a symmetric equation of the plane σ that
More informationChapter 4: Radicals and Complex Numbers
Chapter : Radicals and Complex Numbers Section.1: A Review of the Properties of Exponents #1-: Simplify the expression. 1) x x ) z z ) a a ) b b ) 6) 7) x x x 8) y y y 9) x x y 10) y 8 b 11) b 7 y 1) y
More informationAlgebraic Expressions
Algebraic Expressions 1. Expressions are formed from variables and constants. 2. Terms are added to form expressions. Terms themselves are formed as product of factors. 3. Expressions that contain exactly
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 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 informationChapter 12 Review Vector. MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 1 / 30
Chapter 12 Review Vector MATH 126 (Section 9.5) Vector and Scalar The University of Kansas 1 / 30 iclicker 1: Let v = PQ where P = ( 2, 5) and Q = (1, 2). Which of the following vectors with the given
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 informationGeometry Arcs and Chords. Geometry Mr. Austin
10.2 Arcs and Chords Mr. Austin Objectives/Assignment Use properties of arcs of circles, as applied. Use properties of chords of circles. Assignment: pp. 607-608 #3-47 Reminder Quiz after 10.3 and 10.5
More informationCreated by T. Madas VECTOR OPERATORS. Created by T. Madas
VECTOR OPERATORS GRADIENT gradϕ ϕ Question 1 A surface S is given by the Cartesian equation x 2 2 + y = 25. a) Draw a sketch of S, and describe it geometrically. b) Determine an equation of the tangent
More information26. LECTURE 26. Objectives
6. LECTURE 6 Objectives I understand the idea behind the Method of Lagrange Multipliers. I can use the method of Lagrange Multipliers to maximize a multivariate function subject to a constraint. Suppose
More informationChapter 7: Exponents
Chapter : Exponents Algebra Chapter Notes Name: Algebra Homework: Chapter (Homework is listed by date assigned; homework is due the following class period) HW# Date In-Class Homework M / Review of Sections.-.
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 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 informationb g 6. P 2 4 π b g b g of the way from A to B. LATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME OF THE CHAPTER TEST ON ASSIGNMENT DUE
A Trig/Math Anal Name No LATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME OF THE CHAPTER TEST ON HW NO. SECTIONS (Brown Book) ASSIGNMENT DUE V 1 1 1/1 Practice Set A V 1 3 Practice Set B #1 1 V B 1
More informationDATE: MATH ANALYSIS 2 CHAPTER 12: VECTORS & DETERMINANTS
NAME: PERIOD: DATE: MATH ANALYSIS 2 MR. MELLINA CHAPTER 12: VECTORS & DETERMINANTS Sections: v 12.1 Geometric Representation of Vectors v 12.2 Algebraic Representation of Vectors v 12.3 Vector and Parametric
More information25. Chain Rule. Now, f is a function of t only. Expand by multiplication:
25. Chain Rule The Chain Rule is present in all differentiation. If z = f(x, y) represents a two-variable function, then it is plausible to consider the cases when x and y may be functions of other variable(s).
More informationChapter 10 Exercise 10.1
Chapter 0 Exercise 0. Q.. A(, ), B(,), C(, ), D(, ), E(0,), F(,), G(,0), H(, ) Q.. (i) nd (vi) st (ii) th (iii) nd (iv) rd (v) st (vii) th (viii) rd (ix) st (viii) rd Q.. (i) Y (v) X (ii) Y (vi) X (iii)
More informationQ1. The sum of the lengths of any two sides of a triangle is always (greater/lesser) than the length of the third side. askiitians
Class: VII Subject: Math s Topic: Properties of triangle No. of Questions: 20 Q1. The sum of the lengths of any two sides of a triangle is always (greater/lesser) than the length of the third side. Greater
More informationPractice Problems for the Final Exam
Math 114 Spring 2017 Practice Problems for the Final Exam 1. The planes 3x + 2y + z = 6 and x + y = 2 intersect in a line l. Find the distance from the origin to l. (Answer: 24 3 ) 2. Find the area of
More informationMath 20C Homework 2 Partial Solutions
Math 2C Homework 2 Partial Solutions Problem 1 (12.4.14). Calculate (j k) (j + k). Solution. The basic properties of the cross product are found in Theorem 2 of Section 12.4. From these properties, we
More information10.2,3,4. Vectors in 3D, Dot products and Cross Products
Name: Section: 10.2,3,4. Vectors in 3D, Dot products and Cross Products 1. Sketch the plane parallel to the xy-plane through (2, 4, 2) 2. For the given vectors u and v, evaluate the following expressions.
More informationAdditional Practice Lessons 2.02 and 2.03
Additional Practice Lessons 2.02 and 2.03 1. There are two numbers n that satisfy the following equations. Find both numbers. a. n(n 1) 306 b. n(n 1) 462 c. (n 1)(n) 182 2. The following function is defined
More informationDirectional Derivatives and Gradient Vectors. Suppose we want to find the rate of change of a function z = f x, y at the point in the
14.6 Directional Derivatives and Gradient Vectors 1. Partial Derivates are nice, but they only tell us the rate of change of a function z = f x, y in the i and j direction. What if we are interested in
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 information1 a 2 b 10 c 4 d 12. e 7 f 11 g 12 h 30. i 8 j 7 k 5 l a 9 b 14 c 3 d 10. e 11 f 5 g 15 h 1. i 12 j 2 k 35 l 49.
Cambridge Essentials Mathematics Core 7 A1.1 Answers A1.1 Answers 1 a 2 b 10 c 4 d 12 e 7 f 11 g 12 h 30 i 8 j 7 k 5 l 20 2 a 9 b 14 c 3 d 10 e 11 f 5 g 15 h 1 i 12 j 2 k 35 l 49 m 25 n 1 o 1 p 18 3 a
More informationReview 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 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 informationCHAPTER 10 TRIGONOMETRY
CHAPTER 10 TRIGONOMETRY EXERCISE 39, Page 87 1. Find the length of side x in the diagram below. By Pythagoras, from which, 2 25 x 7 2 x 25 7 and x = 25 7 = 24 m 2. Find the length of side x in the diagram
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 informationReview for Exam 1. (a) Find an equation of the line through the point ( 2, 4, 10) and parallel to the vector
Calculus 3 Lia Vas Review for Exam 1 1. Surfaces. Describe the following surfaces. (a) x + y = 9 (b) x + y + z = 4 (c) z = 1 (d) x + 3y + z = 6 (e) z = x + y (f) z = x + y. Review of Vectors. (a) Let a
More informationGeometry 1 st Semester review Name
Geometry 1 st Semester review Name 1. What are the next three numbers in this sequence? 0, 3, 9, 18, For xercises 2 4, refer to the figure to the right. j k 2. Name the point(s) collinear to points H and
More informationQ4. In ABC, AC = AB and B = 50. Find the value of C. SECTION B. Q5. Find two rational numbers between 1 2 and.
SUMMATIVE ASSESSMENT 1 (2013 2014) CLASS IX (SET I) SUBJECT : MATHEMATICS Time: 3 hours M.M. : 90 General Instructions : (i) All questions are compulsory. (ii) The question paper consists of 31 questions
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 information16.1 Vector Fields. Lukas Geyer. M273, Fall Montana State University. Lukas Geyer (MSU) 16.1 Vector Fields M273, Fall / 16
16.1 Vector Fields Lukas Geyer Montana State University M273, Fall 2011 Lukas Geyer (MSU) 16.1 Vector Fields M273, Fall 2011 1 / 16 Vector Fields Definition An n-dimensional vector field is a function
More informationMATH 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 informationUNC Charlotte Super Competition Level 3 Test March 4, 2019 Test with Solutions for Sponsors
. Find the minimum value of the function f (x) x 2 + (A) 6 (B) 3 6 (C) 4 Solution. We have f (x) x 2 + + x 2 + (D) 3 4, which is equivalent to x 0. x 2 + (E) x 2 +, x R. x 2 + 2 (x 2 + ) 2. How many solutions
More informationNotes for Comp 497 (Comp 454) Week 5 2/22/05. Today we will look at some of the rest of the material in Part 1 of the book.
Notes for Comp 497 (Comp 454) Week 5 2/22/05 Today we will look at some of the rest of the material in Part 1 of the book Errata (Chapter 9) Chapter 9 p. 177, line 11, replace "for L 1 " by "for L 2 ".
More informationIE 5531: Engineering Optimization I
IE 5531: Engineering Optimization I Lecture 12: Nonlinear optimization, continued Prof. John Gunnar Carlsson October 20, 2010 Prof. John Gunnar Carlsson IE 5531: Engineering Optimization I October 20,
More information5-1 Practice Form K. Midsegments of Triangles. Identify three pairs of parallel segments in the diagram.
5-1 Practice Form K Midsegments of Triangles Identify three pairs of parallel segments in the diagram. 1. 2. 3. Name the segment that is parallel to the given segment. 4. MN 5. ON 6. AB 7. CB 8. OM 9.
More information(a 1. By convention the vector a = and so on. r = and b =
By convention the vector a = (a 1 a 3), a and b = (b1 b 3), b and so on. r = ( x z) y There are two sort of half-multiplications for three dimensional vectors. a.b gives an ordinary number (not a vector)
More informationGive a geometric description of the set of points in space whose coordinates satisfy the given pair of equations.
1. Give a geometric description of the set of points in space whose coordinates satisfy the given pair of equations. x + y = 5, z = 4 Choose the correct description. A. The circle with center (0,0, 4)
More informationCHAPTER 4 Stress Transformation
CHAPTER 4 Stress Transformation ANALYSIS OF STRESS For this topic, the stresses to be considered are not on the perpendicular and parallel planes only but also on other inclined planes. A P a a b b P z
More informationParameterization and Vector Fields
Parameterization and Vector Fields 17.1 Parameterized Curves Curves in 2 and 3-space can be represented by parametric equations. Parametric equations have the form x x(t), y y(t) in the plane and x x(t),
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 information16.2 Line Integrals. Lukas Geyer. M273, Fall Montana State University. Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall / 21
16.2 Line Integrals Lukas Geyer Montana State University M273, Fall 211 Lukas Geyer (MSU) 16.2 Line Integrals M273, Fall 211 1 / 21 Scalar Line Integrals Definition f (x) ds = lim { s i } N f (P i ) s
More informationMath 11 Fall 2018 Midterm 1
Math 11 Fall 2018 Midterm 1 October 3, 2018 NAME: SECTION (check one box): Section 1 (I. Petkova 10:10) Section 2 (M. Kobayashi 11:30) Section 3 (W. Lord 12:50) Section 4 (M. Kobayashi 1:10) Instructions:
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 informationMath 103, Review Problems for the First Midterm
Math 0, Review Problems for the First Mierm Ivan Matić. Draw the curve r (t) = cost, sin t, sint and find the tangent line to the curve at t = 0. Find the normal vector to the curve at t = 0.. Find the
More information81-E 2. Ans. : 2. Universal set U = { 2, 3, 5, 6, 10 }, subset A = { 5, 6 }. The diagram which represents A / is. Ans. : ( SPACE FOR ROUGH WORK )
81-E 2 General Instructions : i) The question-cum-answer booklet contains two Parts, Part A & Part B. ii) iii) iv) Part A consists of 60 questions and Part B consists of 16 questions. Space has been provided
More information