VECTORS IN A STRAIGHT LINE
|
|
- Randolph Blake
- 5 years ago
- Views:
Transcription
1 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. It passes through two known fixed points. I. A line with known direction, d, passing through a fixed point d A The diagram shows a straight line passing through a fixed point with a position vector a which is parallel to a given vector d. a r R Let r be the position vector of a point R on the line. O Since AR is parallel to d, then: r a AR = d = d r = a + d Thus every point on the line has a position vector of the form a + d. The equation r = a + d is called the vector equation of the line through A parallel to d. The vector d is a direction vector of the line. Examples :. Find the vector equation of the line which is parallel to the vector i j + 3k and passes through the point 5 4. Hence write down the parametric and Cartesian equations of the line.
2 . Find the vector equation of the line passing through the point with position vector i j + 3k and parallel to the vector j k. Hence write down the parametric and Cartesian equations of the line. II. A line through two fixed points Consider now the line through two points A and B with position vectors a and b respectively. R A B a b r If a point R, with position vector r, lies on the line, then for some scalar λ, 0 Example : AR = λ AB r a = λ b a r = a + λ b a Find the vector equation of the straight line passing through the points A B 3 0 and. Find the parametric and Cartesian equations of the line. Find in each case the coordinates of the points where the line crosses the xy plane, the yz plane and the zx plane.
3 * A general point on the line is the parametric equation of the line. Example : x = + λ, y = 3λ, z = λ At the point where the line crosses the xy plane z = 0 3 Therefore the line crosses the xy plane at the point. Note : Crosses the xy plane Crosses the yz plane Crosses the xz plane B. Pairs of Lines The location of two lines in space may be such that I. The lines are parallel. II. The lines are not parallel and they intersect. III. The lines are not parallel and do not intersect. Such lines are called skew.. Parallel Lines If two lines are parallel, this property can be observed from their equations (such that they will have parallel direction vectors). Example : l : r = + λ 3 l : r = 4 + λ 6
4 . Non-parallel lines r = a + μd Direction ratios (d and d ) are different. Should the lines intersect, there must be unique values λ and μ such that : 4 r = a + λd r = r a + λd = a + μd Example: l : r = λ i + 3λj + λ + 5 k l : r = μi + μ 4 j + 3k If no such values can be found, then the lines do not intersect. 3. Skew Lines The lines are not parallel and do not have a point of intersection. We cannot find values for λ and μ which satisfy all three equations. Example: l : r = + λ i + 3λj + + 4λ k l : r = + 4μ i + 3 μ j + μk
5 C. Angle between a Pair of Lines The angle between two lines l and l is ambiguous as it may be α or 80 α. l But the angle between two vectors a and b is unique. 5 d 80 α It is the angle between their directions when those directions both converge or both diverge from a point. θ d α l If two lines have equations : l : r = a + λd and l : r = a + μd The angle between any two lines depends only on their directions : d. d = d d cos θ If the lines are perpendicular then, Example : Find the angle between the lines r = i j + 3k + λ i 3j + 6k r = i 7j + 0k + μ i + j + k
6 D. Perpendicular Distance from a Point to the Given Line In general P PQ is perpendicular to l or PQ is perpendicular to d. 6 Q d l Examples :. Find the perpendicular distance from a point P, with position vector with vector equation r = 3 + λ to the line l
7 . Find the perpendicular distance (shortest distance) of a point from a point A,, 3 7 to the line x 4 = y = z.
8 E. Vector Equations of a Plane Scalar Form n VECTORS IN A PLANE Consider the plane which contains the point A with position vector a and is perpendicular to n. If r is the positive vector for any point R on the plane. 8 A R Since AR is perpendicular to n. a r O Scalar Form : If r = xi + yj + zk n = Ai + Bj + Ck Cartesian Form Parametric Form q A p R Consider the plane is parallel to vector p and q (p is not parallel to q) and also contains point A with position vector a. If R is any point on the plane: a r O
9 Examples :. Find the vector equation of the plane passing through the point with position vector i j + k and is perpendicular to the vector 3i + j 4k. 9. Write down a vector equation of the plane which contains A,, 3 and is perpendicular to 3i + 4j + 5k in scalar and Cartesian form. Hence show that the point E, 5, 6 does not lie on the plane. 3. Find the vector equation of the plane that contains the three points, 5, 3,, 3, 5 and,, 0. Show that the point 0, 3, 0 lies on the plane.
10 4. Find the Cartesian equation from the vector equation of a plane : r = λ 3 + μ Find the Cartesian equation of the plane that passes through the points,,, 3,, and,,.
11 6. Find the Cartesian equation of the plane through the point, 3, parallel to the plane x + 4y 5z =. F. Intersection of a line and a plane To find where a line meets a plane, we need to find a point on the plane that satisfies the equation for the plane. Steps :. Write down the coordination of a general point on the line.. Use these coordinates in the formula for the equation of the plane. 3. Solve the equation and find the coordinates. Examples :. Find where the line = meets the plane + =.. Find the vector equation of the line passing through the point 3,, and perpendicular to the plane. + = 4. Find also the point of intersection of this line and the plane.
12 G. To Prove that a Line Lies in a Plane Conditions :. The line must be parallel to the plane.. The point through which the line passes must satisfy the plane. Examples :. Show that the line = lies in the plane. 3 + = H. The Angle Between Two Planes The angle between the normals is :. = cos Example : Find the acute angle between two planes whose vector equation are. + = 3 and. + =
13 I. The Angle Between a Line and a Plane 3 Example : Find the angle between the line = and the plane. + = 4 The Line Intersection of Two Planes ( When two planes intersect, they form a line) As the line of intersection of two planes. =. = Is contained in both planes, it is perpendicular to both and.
14 4. Find the line of intersection of these two planes in both Cartesian and Vector equation;. + 3 = 6 Examples :. + = 4
15 . Find in both Cartesian and vector equation of the line of intersection of the two planes: = = 4 5
16 J. The Distance of a Point from a Plane The Distance of a point P from a plane 6. = is. where = And assuming that the point P lies on the plane which is parallel to, then therefore the equation of plane Where distance from the origin =... =. Thus the distance of a point P from a plane is Note :. If P and O (origin) are on the same side of plane, the result will be negative (-ve).. If P and O (origin) are in the opposite side of the plane, the result will be positive (+ve). Example : Find the distance of the point, 3, 3 from a the plane with equation = 9
(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 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 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 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 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 information6.1.1 Angle between Two Lines Intersection of Two lines Shortest Distance from a Point to a Line
CHAPTER 6 : VECTORS 6. Lines in Space 6.. Angle between Two Lines 6.. Intersection of Two lines 6..3 Shortest Distance from a Point to a Line 6. Planes in Space 6.. Intersection of Two Planes 6.. Angle
More 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 information9.5. Lines and Planes. Introduction. Prerequisites. Learning Outcomes
Lines and Planes 9.5 Introduction Vectors are very convenient tools for analysing lines and planes in three dimensions. In this Section you will learn about direction ratios and direction cosines and then
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 information- 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: 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
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 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 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 information- 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 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 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 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 informationDirectional Derivative and the Gradient Operator
Chapter 4 Directional Derivative and the Gradient Operator The equation z = f(x, y) defines a surface in 3 dimensions. We can write this as z f(x, y) = 0, or g(x, y, z) = 0, where g(x, y, z) = z f(x, y).
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 informationVECTORS TEST. 1. Show that the vectors a and b given by a = i + j + k and b = 2i + j 3k are perpendicular. [3]
VECTORS TEST Show that the vectors a and b given by a = i + j + k and b = i + j k are perpendicular [] Find a unit vector which is parallel to the vector a [] Find an equation for the line through the
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 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 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 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 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 informationStudy guide for Exam 1. by William H. Meeks III October 26, 2012
Study guide for Exam 1. by William H. Meeks III October 2, 2012 1 Basics. First we cover the basic definitions and then we go over related problems. Note that the material for the actual midterm may include
More informationVECTORS IN COMPONENT FORM
VECTORS IN COMPONENT FORM In Cartesian coordinates any D vector a can be written as a = a x i + a y j + a z k a x a y a x a y a z a z where i, j and k are unit vectors in x, y and z directions. i = j =
More 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 informationRegent College. Maths Department. Core Mathematics 4. Vectors
Regent College Maths Department Core Mathematics 4 Vectors Page 1 Vectors By the end of this unit you should be able to find: a unit vector in the direction of a. the distance between two points (x 1,
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 informationCourse 2BA1: Hilary Term 2007 Section 8: Quaternions and Rotations
Course BA1: Hilary Term 007 Section 8: Quaternions and Rotations David R. Wilkins Copyright c David R. Wilkins 005 Contents 8 Quaternions and Rotations 1 8.1 Quaternions............................ 1 8.
More informationCHAPTER 10 VECTORS POINTS TO REMEMBER
For more important questions visit : www4onocom CHAPTER 10 VECTORS POINTS TO REMEMBER A quantity that has magnitude as well as direction is called a vector It is denoted by a directed line segment Two
More informationChapter 2 - Vector Algebra
A spatial vector, or simply vector, is a concept characterized by a magnitude and a direction, and which sums with other vectors according to the Parallelogram Law. A vector can be thought of as an arrow
More informationCreated by T. Madas SURFACE INTEGRALS. Created by T. Madas
SURFACE INTEGRALS Question 1 Find the area of the plane with equation x + 3y + 6z = 60, 0 x 4, 0 y 6. 8 Question A surface has Cartesian equation y z x + + = 1. 4 5 Determine the area of the surface which
More informationthe coordinates of C (3) Find the size of the angle ACB. Give your answer in degrees to 2 decimal places. (4)
. The line l has equation, 2 4 3 2 + = λ r where λ is a scalar parameter. The line l 2 has equation, 2 0 5 3 9 0 + = µ r where μ is a scalar parameter. Given that l and l 2 meet at the point C, find the
More informationMath 234. What you should know on day one. August 28, You should be able to use general principles like. x = cos t, y = sin t, 0 t π.
Math 234 What you should know on day one August 28, 2001 1 You should be able to use general principles like Length = ds, Area = da, Volume = dv For example the length of the semi circle x = cos t, y =
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 information( ) = ( ) ( ) = ( ) = + = = = ( ) Therefore: , where t. Note: If we start with the condition BM = tab, we will have BM = ( x + 2, y + 3, z 5)
Chapter Exercise a) AB OB OA ( xb xa, yb ya, zb za),,, 0, b) AB OB OA ( xb xa, yb ya, zb za) ( ), ( ),, 0, c) AB OB OA x x, y y, z z (, ( ), ) (,, ) ( ) B A B A B A ( ) d) AB OB OA ( xb xa, yb ya, zb za)
More informationDirectional Derivatives in the Plane
Directional Derivatives in the Plane P. Sam Johnson April 10, 2017 P. Sam Johnson (NIT Karnataka) Directional Derivatives in the Plane April 10, 2017 1 / 30 Directional Derivatives in the Plane Let z =
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 informationVectors. J.R. Wilson. September 28, 2017
Vectors J.R. Wilson September 28, 2017 This chapter introduces vectors that are used in many areas of physics (needed for classical physics this year). One complication is that a number of different forms
More informationVectors are used to represent quantities such as force and velocity which have both. and. The magnitude of a vector corresponds to its.
Fry Texas A&M University Math 150 Chapter 9 Fall 2014 1 Chapter 9 -- Vectors Remember that is the set of real numbers, often represented by the number line, 2 is the notation for the 2-dimensional plane.
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 information9 Mixed Exercise. vector equation is. 4 a
9 Mixed Exercise a AB r i j k j k c OA AB 7 i j 7 k A7,, and B,,8 8 AB 6 A vector equation is 7 r x 7 y z (i j k) j k a x y z a a 7, Pearson Education Ltd 7. Copying permitted for purchasing institution
More information5 Find an equation of the circle in which AB is a diameter in each case. a A (1, 2) B (3, 2) b A ( 7, 2) B (1, 8) c A (1, 1) B (4, 0)
C2 CRDINATE GEMETRY Worksheet A 1 Write down an equation of the circle with the given centre and radius in each case. a centre (0, 0) radius 5 b centre (1, 3) radius 2 c centre (4, 6) radius 1 1 d centre
More information1.1 Bound and Free Vectors. 1.2 Vector Operations
1 Vectors Vectors are used when both the magnitude and the direction of some physical quantity are required. Examples of such quantities are velocity, acceleration, force, electric and magnetic fields.
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 informationUnit 8. ANALYTIC GEOMETRY.
Unit 8. ANALYTIC GEOMETRY. 1. VECTORS IN THE PLANE A vector is a line segment running from point A (tail) to point B (head). 1.1 DIRECTION OF A VECTOR The direction of a vector is the direction of the
More information12 th Class Mathematics Paper
th Class Mathematics Paper Maimum Time: hours Maimum Marks: 00 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 9 questions divided into four sections A, B, C
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 informationVECTOR ALGEBRA. 3. write a linear vector in the direction of the sum of the vector a = 2i + 2j 5k and
1 mark questions VECTOR ALGEBRA 1. Find a vector in the direction of vector 2i 3j + 6k which has magnitude 21 units Ans. 6i-9j+18k 2. Find a vector a of magnitude 5 2, making an angle of π with X- axis,
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 2/13/13, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.2. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241sp13/241.html)
More informationVectors are used to represent quantities such as force and velocity which have both. and. The magnitude of a vector corresponds to its.
Fry Texas A&M University Fall 2016 Math 150 Notes Chapter 9 Page 248 Chapter 9 -- Vectors Remember that is the set of real numbers, often represented by the number line, 2 is the notation for the 2-dimensional
More informationMathematics 13: Lecture 4
Mathematics 13: Lecture Planes Dan Sloughter Furman University January 10, 2008 Dan Sloughter (Furman University) Mathematics 13: Lecture January 10, 2008 1 / 10 Planes in R n Suppose v and w are nonzero
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 informationa b = a a a and that has been used here. ( )
Review Eercise ( i j+ k) ( i+ j k) i j k = = i j+ k (( ) ( ) ) (( ) ( ) ) (( ) ( ) ) = i j+ k = ( ) i ( ( )) j+ ( ) k = j k Hence ( ) ( i j+ k) ( i+ j k) = ( ) + ( ) = 8 = Formulae for finding the vector
More informationExercises 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 information1. The unit vector perpendicular to both the lines. Ans:, (2)
1. The unit vector perpendicular to both the lines x 1 y 2 z 1 x 2 y 2 z 3 and 3 1 2 1 2 3 i 7j 7k i 7j 5k 99 5 3 1) 2) i 7j 5k 7i 7j k 3) 4) 5 3 99 i 7j 5k Ans:, (2) 5 3 is Solution: Consider i j k a
More information9. Stress Transformation
9.7 ABSOLUTE MAXIMUM SHEAR STRESS A pt in a body subjected to a general 3-D state of stress will have a normal stress and shear-stress components acting on each of its faces. We can develop stress-transformation
More informationIf you must be wrong, how little wrong can you be?
MATH 2411 - Harrell If you must be wrong, how little wrong can you be? Lecture 13 Copyright 2013 by Evans M. Harrell II. About the test Median was 35, range 25 to 40. As it is written: About the test Percentiles:
More informationPage Problem Score Max Score a 8 12b a b 10 14c 6 6
Fall 14 MTH 34 FINAL EXAM December 8, 14 Name: PID: Section: Instructor: DO NOT WRITE BELOW THIS LINE. Go to the next page. Page Problem Score Max Score 1 5 5 1 3 5 4 5 5 5 6 5 7 5 8 5 9 5 1 5 11 1 3 1a
More informationProf. B V S Viswanadham, Department of Civil Engineering, IIT Bombay
50 Module 4: Lecture 1 on Stress-strain relationship and Shear strength of soils Contents Stress state, Mohr s circle analysis and Pole, Principal stressspace, Stress pathsin p-q space; Mohr-Coulomb failure
More informationEngineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Kinematics
Engineering Mechanics Prof. U. S. Dixit Department of Mechanical Engineering Indian Institute of Technology, Guwahati Kinematics Module 10 - Lecture 24 Kinematics of a particle moving on a curve Today,
More informationVectors. J.R. Wilson. September 27, 2018
Vectors J.R. Wilson September 27, 2018 This chapter introduces vectors that are used in many areas of physics (needed for classical physics this year). One complication is that a number of different forms
More informationVector Calculus - GATE Study Material in PDF
Vector Calculus - GATE Study Material in PDF In previous articles, we have already seen the basics of Calculus Differentiation and Integration and applications. In GATE 2018 Study Notes, we will be introduced
More informationVectors in the new Syllabus. Taylors College, Sydney. MANSW Conference Sept. 16, 2017
Vectors in the new Syllabus by Derek Buchanan Taylors College, Sydney MANSW Conference Sept. 6, 07 Quantities which have only magnitude are called scalars. Quantities which have magnitude and direction
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 VECTORS II. Triple products 2. Differentiation and integration of vectors 3. Equation of a line 4. Equation of a plane.
More information9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b
vectors and POLAR COORDINATES LEARNING OBJECTIVES In this section, ou will: View vectors geometricall. Find magnitude and direction. Perform vector addition and scalar multiplication. Find the component
More informationBSc (Hons) in Computer Games Development. vi Calculate the components a, b and c of a non-zero vector that is orthogonal to
1 APPLIED MATHEMATICS INSTRUCTIONS Full marks will be awarded for the correct solutions to ANY FIVE QUESTIONS. This paper will be marked out of a TOTAL MAXIMUM MARK OF 100. Credit will be given for clearly
More informationUNIT 1 VECTORS INTRODUCTION 1.1 OBJECTIVES. Stucture
UNIT 1 VECTORS 1 Stucture 1.0 Introduction 1.1 Objectives 1.2 Vectors and Scalars 1.3 Components of a Vector 1.4 Section Formula 1.5 nswers to Check Your Progress 1.6 Summary 1.0 INTRODUCTION In this unit,
More informationQuestions. Exercise (1)
Questions Exercise (1) (1) hoose the correct answer: 1) The acute angle supplements. angle. a) acute b) obtuse c) right d) reflex 2) The right angle complements angle whose measure is. a) 0 b) 45 c) 90
More informationName: ID: Math 233 Exam 1. Page 1
Page 1 Name: ID: This exam has 20 multiple choice questions, worth 5 points each. You are allowed to use a scientific calculator and a 3 5 inch note card. 1. Which of the following pairs of vectors are
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 informationChapter 6: Vector Analysis
Chapter 6: Vector Analysis We use derivatives and various products of vectors in all areas of physics. For example, Newton s 2nd law is F = m d2 r. In electricity dt 2 and magnetism, we need surface and
More informationMATH2000 Flux integrals and Gauss divergence theorem (solutions)
DEPARTMENT O MATHEMATIC MATH lux integrals and Gauss divergence theorem (solutions ( The hemisphere can be represented as We have by direct calculation in terms of spherical coordinates. = {(r, θ, φ r,
More information(1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3
Math 127 Introduction and Review (1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3 MATH 127 Introduction to Calculus III
More informationCongruence Axioms. Data Required for Solving Oblique Triangles
Math 335 Trigonometry Sec 7.1: Oblique Triangles and the Law of Sines In section 2.4, we solved right triangles. We now extend the concept to all triangles. Congruence Axioms Side-Angle-Side SAS Angle-Side-Angle
More informationVector Fields and Line Integrals The Fundamental Theorem for Line Integrals
Math 280 Calculus III Chapter 16 Sections: 16.1, 16.2 16.3 16.4 16.5 Topics: Vector Fields and Line Integrals The Fundamental Theorem for Line Integrals Green s Theorem Curl and Divergence Section 16.1
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 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 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 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 informationPossible C4 questions from past papers P1 P3
Possible C4 questions from past papers P1 P3 Source of the original question is given in brackets, e.g. [P January 001 Question 1]; a question which has been edited is indicated with an asterisk, e.g.
More informationProblem Solving 1: Line Integrals and Surface Integrals
A. Line Integrals MASSACHUSETTS INSTITUTE OF TECHNOLOY Department of Physics Problem Solving 1: Line Integrals and Surface Integrals The line integral of a scalar function f ( xyz),, along a path C is
More informationUnit #17: Spring Trig Unit. A. First Quadrant Notice how the x-values decrease by while the y-values increase by that same amount.
Name Unit #17: Spring Trig Unit Notes #1: Basic Trig Review I. Unit Circle A circle with center point and radius. A. First Quadrant Notice how the x-values decrease by while the y-values increase by that
More informationCreated by T. Madas VECTOR MOMENTS. Created by T. Madas
VECTOR MOMENTS Question 1 (**) The vectors i, j and k are unit vectors mutually perpendicular to one another. Relative to a fixed origin O, a light rigid rod has its ends located at the points 0, 7,4 B
More informationName: ID: Math 233 Exam 1. Page 1
Page 1 Name: ID: This exam has 20 multiple choice questions, worth 5 points each. You are allowed to use a scientific calculator and a 3 5 inch note card. 1. Which of the following pairs of vectors are
More informationTABLE OF CONTENTS 2 CHAPTER 1
TABLE OF CONTENTS CHAPTER 1 Quadratics CHAPTER Functions 3 CHAPTER 3 Coordinate Geometry 3 CHAPTER 4 Circular Measure 4 CHAPTER 5 Trigonometry 4 CHAPTER 6 Vectors 5 CHAPTER 7 Series 6 CHAPTER 8 Differentiation
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 information11.4 Dot Product Contemporary Calculus 1
11.4 Dot Product Contemporary Calculus 1 11.4 DOT PRODUCT In the previous sections we looked at the meaning of vectors in two and three dimensions, but the only operations we used were addition and subtraction
More informationContents. MATH 32B-2 (18W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables. 1 Multiple Integrals 3. 2 Vector Fields 9
MATH 32B-2 (8W) (L) G. Liu / (TA) A. Zhou Calculus of Several Variables Contents Multiple Integrals 3 2 Vector Fields 9 3 Line and Surface Integrals 5 4 The Classical Integral Theorems 9 MATH 32B-2 (8W)
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 informationGroup, Rings, and Fields Rahul Pandharipande. I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S,
Group, Rings, and Fields Rahul Pandharipande I. Sets Let S be a set. The Cartesian product S S is the set of ordered pairs of elements of S, A binary operation φ is a function, S S = {(x, y) x, y S}. φ
More informationPRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives.
PRACTICE PROBLEMS Please let me know if you find any mistakes in the text so that i can fix them. 1.1. Let Show that f is C 1 and yet How is that possible? 1. Mixed partial derivatives f(x, y) = {xy x
More informationCourse MA2C02, Hilary Term 2010 Section 4: Vectors and Quaternions
Course MA2C02, Hilary Term 2010 Section 4: Vectors and Quaternions David R. Wilkins Copyright c David R. Wilkins 2000 2010 Contents 4 Vectors and Quaternions 47 4.1 Vectors...............................
More informationMATRICES EXAM QUESTIONS
MATRICES EXAM QUESTIONS (Part One) Question 1 (**) The matrices A, B and C are given below in terms of the scalar constants a, b, c and d, by 2 3 A =, 1 a b 1 B =, 2 4 1 c C =. d 4 Given that A + B = C,
More information4.1 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 informationContents. 1 Vectors, Lines and Planes 1. 2 Gaussian Elimination Matrices Vector Spaces and Subspaces 124
Matrices Math 220 Copyright 2016 Pinaki Das This document is freely redistributable under the terms of the GNU Free Documentation License For more information, visit http://wwwgnuorg/copyleft/fdlhtml Contents
More information