Overview. Distances in R 3. Distance from a point to a plane. Question

Size: px
Start display at page:

Download "Overview. Distances in R 3. Distance from a point to a plane. Question"

Transcription

1 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. n (r ) 0 The vector equation for a plane describes arbitrar points r in terms of a specific point and the normal vector n. Question How 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? (From Stewart 10.5) Dr Scott Morrison (ANU) MATH1014 Notes Second Semester / 17 Distances in R 3 The distance between two points is the length of the line segment connecting them. However, there s more than one line segment from a point P to a line L, so what do we mean b the distance between them? The distance between an two subsets A, B of R 3 is the smallest distance between points a and b, where a is in A and b is in B. To determine the distance between a point P and a line L, we need to find the point Q on L which is closest to P, and then measure the length of the line segment PQ. This line segment is orthogonal to L. To determine the distance between a point P and a plane S, we need to find the point Q on S which is closest to P, and then measture the length of the line segment PQ. Again, this line segment is orthogonal to S. In both cases, the ke to computing these distances is drawing a picture and using one of the vector product identitites. Dr Scott Morrison (ANU) MATH1014 Notes Second Semester / 17 Distance from a point to a plane We find a formula for the distance s from a point P 1 ( 1, 1, 1 ) to the plane A + B + C + D 0. P 1 s n b r P 0 Let P 0 ( 0, 0, 0 ) be an point in the given plane and let b be the vector corresponding to P 0 P 1. Then b 1 0, 1 0, 1 0. The distance s from P 1 to the plane is equal to the absolute value of the scalar projection of b onto the normal vector n A, B, C. Dr Scott Morrison (ANU) MATH1014 Notes Second Semester / 17

2 s comp n b n b n A( 1 0 ) + B( 1 0 ) + C( 1 0 ) A 1 + B 1 + C 1 (A 0 + B 0 + C 0 ) Since P 0 is on the plane, its coordinates satisf the equation of the plane and so we have A 0 + B 0 + C 0 + D 0. Thus the formula for s can be written s A 1 + B 1 + C 1 + D Dr Scott Morrison (ANU) MATH1014 Notes Second Semester / 17 Eample 1 We find the distance from the point (1, 2, 0) to the plane From the result above, the distance s is given b where ( 0, 0, 0 ) (1, 2, 0), s A 0 + B 0 + C 0 + D A 3, B 4, C 5 and D 2. This gives s ( 4) 2 + ( 5) ( 4) 2 + ( 5) Dr Scott Morrison (ANU) MATH1014 Notes Second Semester / 17 Distance from a point to a line Question Given a point P 0 ( 0, 0, 0 ) and a line L in R 3, what is the distance from P 0 to L? Tools: describe L using vectors u v u v sin θ Dr Scott Morrison (ANU) MATH1014 Notes Second Semester / 17

3 Distance from a point to a line Let P 0 ( 0, 0, 0 ) and let L be the line through P 1 and parallel to the nonero vector v. Let and r 1 be the position vectors of P 0 and P 1 respectivel. P 2 on L is the point closest to P 0 if and onl if the vector P 2 P 0 is perpendicular to L. vvvv L v P P 2 1 θ -r r 1 s 1 P 0 The distance from P 0 to L is given b s P 2 P 0 P 1 P 0 sin θ r 1 sin θ where θ is the angle between r 1 and v Dr Scott Morrison (ANU) MATH1014 Notes Second Semester / 17 Since we get the formula ( r 1 ) v r 1 v sin θ s r 1 sin θ ( r 1 ) v v Dr Scott Morrison (ANU) MATH1014 Notes Second Semester / 17 Eample 2 Find the distance from the point (1, 1, 1) to the line of intersection of the planes + + 1, The direction of the line is given b v n 1 n 2 where n 1 i + j + k, and n 2 2i j 5k. v n 1 n 2 4i + 7j 3k. P 1 (1,-1/4,1/4) v P 2 -r 1 s P 0 (1,1,-1) In the diagram, P 1 is an arbitrar point on the line. To find such a point, put 1 in the first equation. This gives which can be used in the second equation to find 1/4, and hence 1/4. Dr Scott Morrison (ANU) MATH1014 Notes Second Semester / 17

4 Here P 1 P 0 r j 5 4k. So s ( r 1 ) v v ( 5 4 j 5 4k) ( 4i + 7j 3k) ( 4) ( 3) 2 5i + 5j + 5k Dr Scott Morrison (ANU) MATH1014 Notes Second Semester / 17 Distance between two lines Let L 1 and L 2 be two lines in R 3 such that - L 1 passes through the point P 1 and is parallel to the vector v 1 - L 2 passes through the point P 2 and is parallel to the vector v 2. Let r 1 and r 2 be the position vectors of P 1 and P 2 respectivel. Then parametric equation for these lines are L 1 r r 1 + tv 1 L 2 r r 2 + sv 2 Note that r 2 r 1 P 1 P 2. We want to compute the smallest distance d (simpl called the distance) between the two lines. If the two lines intersect, then d 0. If the two lines do not intersect we can distinguish two cases. Dr Scott Morrison (ANU) MATH1014 Notes Second Semester / 17 Case 1: L 1 and L 2 are parallel and do not intersect. In this case the distance d is simpl the distance from the point P 2 to the line L 1 and is given b d P 1 P 2 v 1 (r 2 r 1 ) v 1 v 1 v 1 Dr Scott Morrison (ANU) MATH1014 Notes Second Semester / 17

5 Case 2: L 1 and L 2 are skew lines. If P 3 and P 4 (with position vectors r 3 and r 4 respectivel) are the points on L 1 and L 2 that are closest to one another, then the vector P 3 P 4 is perpendicular to both lines (i.e. to both v 1 and v 2 ) and therefore parallel to v 1 v 2. The distance d is the length of P 3 P 4. Now P 3 P 4 r 4 r 3 is the vector projection of P 1 P 2 r 2 r 1 along v 1 v 2. Thus the distance d is the absolute value of the scalar projection of r 2 r 1 along v 1 v 2 d r 4 r 3 (r 2 r 1 ) (v 1 v 2 ) v 1 v 2 Observe that if the two lines are parallel then v 1 and v 2 are proportional and thus v 1 v 2 0 (the ero vector) and the above formula does not make sense. Dr Scott Morrison (ANU) MATH1014 Notes Second Semester / 17 Dr Scott Morrison (ANU) MATH1014 Notes Second Semester / 17 Eample 3 Find the distance between the skew lines { and { vvvv v P 1 P 3 1 L 1 r 2 -r 1 L 2 P 4 v 2 P 2 v 1 v 2 We can take P 1 (1, 1, 1), a point on the first line, and P 2 (1, 2, 3) a point on the second line. This gives r 2 r 1 j + 2k. Dr Scott Morrison (ANU) MATH1014 Notes Second Semester / 17

6 Now we need to find v 1 and v 2 : and This gives v 1 (i + 2j) (j + 2k) 4i 2j + k, v 2 (i + j + k) (i 2k) 2i + 3j k. v 1 v 2 i + 2j + 8k. The required distance d is the length of the projection of r 2 r 1 in the direction of v 1 v 2, and is given b d (r 2 r 1 ) (v 1 v 2 ) v 1 v 2 (j + 2k) ( i + 2j + 8k) ( 1) Dr Scott Morrison (ANU) MATH1014 Notes Second Semester / 17

How 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?

How 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 information

12.5 Equations of Lines and Planes

12.5 Equations of Lines and Planes 12.5 Equations of Lines and Planes Equation of Lines Vector Equation of Lines Parametric Equation of Lines Symmetric Equation of Lines Relation Between Two Lines Equations of Planes Vector Equation of

More information

Mathematics 2203, Test 1 - Solutions

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

More information

VECTORS AND THE GEOMETRY OF SPACE

VECTORS 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

Lecture 5. Equations of Lines and Planes. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.

Lecture 5. Equations of Lines and Planes. Dan Nichols MATH 233, Spring 2018 University of Massachusetts. Lecture 5 Equations of Lines and Planes Dan Nichols nichols@math.umass.edu MATH 233, Spring 2018 Universit of Massachusetts Februar 6, 2018 (2) Upcoming midterm eam First midterm: Wednesda Feb. 21, 7:00-9:00

More information

Kevin James. MTHSC 206 Section 12.5 Equations of Lines and Planes

Kevin 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 information

MATH 19520/51 Class 2

MATH 19520/51 Class 2 MATH 19520/51 Class 2 Minh-Tam Trinh University of Chicago 2017-09-27 1 Review dot product. 2 Angles between vectors and orthogonality. 3 Projection of one vector onto another. 4 Cross product and its

More information

Math 241, Exam 1 Information.

Math 241, Exam 1 Information. Math 241, Exam 1 Information. 2/13/13, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.2. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241sp13/241.html)

More information

Overview. Motivation for the inner product. Question. Definition

Overview. Motivation for the inner product. Question. Definition Overview Last time we studied the evolution of a discrete linear dynamical system, and today we begin the final topic of the course (loosely speaking) Today we ll recall the definition and properties of

More information

Department of Mathematical and Statistical Sciences University of Alberta

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

More information

12. Lines in R 3. through P 0 and is parallel to v is written parametrically as a function of t: Using vector notation, the same line is written

12. Lines in R 3. through P 0 and is parallel to v is written parametrically as a function of t: Using vector notation, the same line is written 12. Lines in R 3 Given a point P 0 = (x 0, y 0, z 0 ) and a direction vector v 1 = a, b, c in R 3, a line L that passes through P 0 and is parallel to v is written parametrically as a function of t: x(t)

More information

OA = 6, 5, 4 2, 7, 1 = 4, 12, 3 AB, AC, and = 5, 3, 3 3, 4, 12 = = 39 = 39. AC, AD) = = 3 AC) AC) AD

OA = 6, 5, 4 2, 7, 1 = 4, 12, 3 AB, AC, and = 5, 3, 3 3, 4, 12 = = 39 = 39. AC, AD) = = 3 AC) AC) AD Math Fall First Eam Solutions October, Problem. ( pts.) Consider the points A(, 7, ), B(6,, 4), C(6,, 3), and D(7, 4, 4). (a pts.) Find the area of the parallelogram defined b AB and AC. First we calculate

More information

Geometry review, part I

Geometry review, part I Geometr reie, part I Geometr reie I Vectors and points points and ectors Geometric s. coordinate-based (algebraic) approach operations on ectors and points Lines implicit and parametric equations intersections,

More information

Math 3c Solutions: Exam 1 Fall Graph by eliiminating the parameter; be sure to write the equation you get when you eliminate the parameter.

Math 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 information

Topic 5.2: Introduction to Vector Fields

Topic 5.2: Introduction to Vector Fields Math 75 Notes Topic 5.: Introduction to Vector Fields Tetbook Section: 16.1 From the Toolbo (what you need from previous classes): Know what a vector is. Be able to sketch a vector using its component

More information

Review: critical point or equivalently f a,

Review: critical point or equivalently f a, Review: a b f f a b f a b critical point or equivalentl f a, b A point, is called a of if,, 0 A local ma or local min must be a critical point (but not conversel) 0 D iscriminant (or Hessian) f f D f f

More information

Analytic Geometry in Three Dimensions

Analytic Geometry in Three Dimensions Analtic Geometr in Three Dimensions. The Three-Dimensional Coordinate Sstem. Vectors in Space. The Cross Product of Two Vectors. Lines and Planes in Space The three-dimensional coordinate sstem is used

More information

Three-Dimensional Space; Vectors

Three-Dimensional Space; Vectors Chapter 3 Three-Dimensional Space; Vectors 3.1 Rectangular Coordinates in 3-Space; Spheres Rectangular Coordinate Sstems To begin, consider three mutuall perpendicular coordinate lines, called the -ais,

More information

MATH 151 Engineering Mathematics I

MATH 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 information

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

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

More information

Directional derivatives and gradient vectors (Sect. 14.5) Directional derivative of functions of two variables.

Directional derivatives and gradient vectors (Sect. 14.5) Directional derivative of functions of two variables. Directional derivatives and gradient vectors (Sect. 14.5) Directional derivative of functions of two variables. Partial derivatives and directional derivatives. Directional derivative of functions of three

More information

9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b

9.1 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 information

Multiple Choice. 1.(6 pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.

Multiple Choice. 1.(6 pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9. Multiple Choice.(6 pts) Find smmetric equations of the line L passing through the point (, 5, ) and perpendicular to the plane x + 3 z = 9. (a) x = + 5 3 = z (c) (x ) + 3( 3) (z ) = 9 (d) (e) x = 3 5 =

More information

VECTORS IN A STRAIGHT LINE

VECTORS 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 information

SOLUTIONS TO SECOND PRACTICE EXAM Math 21a, Spring 2003

SOLUTIONS TO SECOND PRACTICE EXAM Math 21a, Spring 2003 SOLUTIONS TO SECOND PRACTICE EXAM Math a, Spring 3 Problem ) ( points) Circle for each of the questions the correct letter. No justifications are needed. Your score will be C W where C is the number of

More information

MATH H53 : Mid-Term-1

MATH 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 information

What you will learn today

What 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 information

Directional derivatives and gradient vectors (Sect. 14.5). Directional derivative of functions of two variables.

Directional derivatives and gradient vectors (Sect. 14.5). Directional derivative of functions of two variables. Directional derivatives and gradient vectors (Sect. 14.5). Directional derivative of functions of two variables. Partial derivatives and directional derivatives. Directional derivative of functions of

More information

Math 323 Exam 2 - Practice Problem Solutions. 2. Given the vectors a = 1,2,0, b = 1,0,2, and c = 0,1,1, compute the following:

Math 323 Exam 2 - Practice Problem Solutions. 2. Given the vectors a = 1,2,0, b = 1,0,2, and c = 0,1,1, compute the following: Math 323 Eam 2 - Practice Problem Solutions 1. Given the vectors a = 2,, 1, b = 3, 2,4, and c = 1, 4,, compute the following: (a) A unit vector in the direction of c. u = c c = 1, 4, 1 4 =,, 1+16+ 17 17

More information

MATH 255 Applied Honors Calculus III Winter Midterm 1 Review Solutions

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

More information

- 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.

- 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 information

Vectors. Section 3: Using the vector product

Vectors. 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 information

(6, 4, 0) = (3, 2, 0). Find the equation of the sphere that has the line segment from P to Q as a diameter.

(6, 4, 0) = (3, 2, 0). Find the equation of the sphere that has the line segment from P to Q as a diameter. Solutions Review for Eam #1 Math 1260 1. Consider the points P = (2, 5, 1) and Q = (4, 1, 1). (a) Find the distance from P to Q. Solution. dist(p, Q) = (4 2) 2 + (1 + 5) 2 + (1 + 1) 2 = 4 + 36 + 4 = 44

More information

Vector Fields. Field (II) Field (V)

Vector Fields. Field (II) Field (V) Math 1a Vector Fields 1. Match the following vector fields to the pictures, below. Eplain our reasoning. (Notice that in some of the pictures all of the vectors have been uniforml scaled so that the picture

More information

problem score possible Total 60

problem score possible Total 60 Math 32 A: Midterm 1, Oct. 24, 2008 Name: ID Number: Section: problem score possible 1. 10 2. 10 3. 10 4. 10 5. 10 6. 10 Total 60 1 1. Determine whether the following points are coplanar: a. P = (8, 14,

More information

MATH1014. Semester 1 Administrative Overview. Neil Montgomery calculus

MATH1014. Semester 1 Administrative Overview. Neil Montgomery calculus MATH1014 Semester 1 Administrative Overview Lecturers: Scott Morrison linear algebra scott.morrison@anu.edu.au Neil Montgomery calculus neil.montgomery@anu.edu.au Dr Scott Morrison (ANU) MATH1014 Notes

More information

MAC Module 5 Vectors in 2-Space and 3-Space II

MAC Module 5 Vectors in 2-Space and 3-Space II MAC 2103 Module 5 Vectors in 2-Space and 3-Space II 1 Learning Objectives Upon completing this module, you should be able to: 1. Determine the cross product of a vector in R 3. 2. Determine a scalar triple

More information

Distances 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: 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 information

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

Study guide for Exam 1. by William H. Meeks III October 26, 2012 Study guide for Exam 1. by William H. Meeks III October 2, 2012 1 Basics. First we cover the basic definitions and then we go over related problems. Note that the material for the actual midterm may include

More information

1 Vectors. c Kun Wang. Math 151, Fall Vector Supplement

1 Vectors. c Kun Wang. Math 151, Fall Vector Supplement Vector Supplement 1 Vectors A vector is a quantity that has both magnitude and direction. Vectors are drawn as directed line segments and are denoted by boldface letters a or by a. The magnitude of a vector

More information

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

(1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3 Math 127 Introduction and Review (1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3 MATH 127 Introduction to Calculus III

More information

MATH 2203 Exam 1 January 26, 2004 S. F. Ellermeyer Name

MATH 2203 Exam 1 January 26, 2004 S. F. Ellermeyer Name MATH 2203 Exam 1 January 26, 2004 S. F. Ellermeyer Name Instructions. This exam contains seven problems, but only six of them will be graded. You maychooseanysixtodo. PleasewriteDON TGRADEontheonethatyoudon

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.

- 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 information

CALCULUS III. prepared by

CALCULUS III. prepared by EXAMINATION 1 CALCULUS III REVIEW PROBLEMS prepared by Antony Foster Department of Mathematics (office: NAC 6-73) The City College of The City University of New York Convent Avenue At 138th Street New

More information

MA123, Chapter 1: Equations, functions and graphs (pp. 1-15)

MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) MA123, Chapter 1: Equations, functions and graphs (pp. 1-15) Date: Chapter Goals: Identif solutions to an equation. Solve an equation for one variable in terms of another. What is a function? Understand

More information

From Lay, 5.4. If we always treat a matrix as defining a linear transformation, what role does diagonalisation play?

From Lay, 5.4. If we always treat a matrix as defining a linear transformation, what role does diagonalisation play? Overview Last week introduced the important Diagonalisation Theorem: An n n matrix A is diagonalisable if and only if there is a basis for R n consisting of eigenvectors of A. This week we ll continue

More information

Introduction to 3D Game Programming with DirectX 9.0c: A Shader Approach

Introduction to 3D Game Programming with DirectX 9.0c: A Shader Approach Introduction to 3D Game Programming with DirectX 90c: A Shader Approach Part I Solutions Note : Please email to frank@moon-labscom if ou find an errors Note : Use onl after ou have tried, and struggled

More information

Review of Prerequisite Skills, p. 350 C( 2, 0, 1) B( 3, 2, 0) y A(0, 1, 0) D(0, 2, 3) j! k! 2k! Section 7.1, pp

Review of Prerequisite Skills, p. 350 C( 2, 0, 1) B( 3, 2, 0) y A(0, 1, 0) D(0, 2, 3) j! k! 2k! Section 7.1, pp . 5. a. a a b a a b. Case If and are collinear, then b is also collinear with both and. But is perpendicular to and c c c b 9 b c, so a a b b is perpendicular to. Case If b and c b c are not collinear,

More information

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

Vector equations of lines in the plane and 3-space (uses vector addition & scalar multiplication). Boise State Math 275 (Ultman) Worksheet 1.6: Lines and Planes From the Toolbox (what you need from previous classes) Plotting points, sketching vectors. Be able to find the component form a vector given

More information

FINDING THE INTERSECTION OF TWO LINES

FINDING 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 information

Mathematics 13: Lecture 4

Mathematics 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 information

Calculus 1 for AE (WI1421LR) Lecture 1: 12.3 The dot product 12.4 The cross product

Calculus 1 for AE (WI1421LR) Lecture 1: 12.3 The dot product 12.4 The cross product Calculus 1 for AE (WI1421LR) : 12.3 The dot product 12.4 The cross product About this course Textbook James Stewart, Calculus, Early Transcendentals, 7th edition Course Coordinator Dr. Roelof Koekoek,

More information

Review of Coordinate Systems

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

More information

R n : The Cross Product

R n : The Cross Product A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler R n : The Cross Product Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION Attribution-NonCommercial-ShareAlike (CC BY-NC-SA)

More information

STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs

STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE. Functions & Graphs STUDY KNOWHOW PROGRAM STUDY AND LEARNING CENTRE Functions & Graphs Contents Functions and Relations... 1 Interval Notation... 3 Graphs: Linear Functions... 5 Lines and Gradients... 7 Graphs: Quadratic

More information

Chapter 2: Vector Geometry

Chapter 2: Vector Geometry Chapter 2: Vector Geometry Daniel Chan UNSW Semester 1 2018 Daniel Chan (UNSW) Chapter 2: Vector Geometry Semester 1 2018 1 / 32 Goals of this chapter In this chapter, we will answer the following geometric

More information

Introduction to Differential Equations

Introduction to Differential Equations Math0 Lecture # Introduction to Differential Equations Basic definitions Definition : (What is a DE?) A differential equation (DE) is an equation that involves some of the derivatives (or differentials)

More information

The geometry of least squares

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

More information

Circles MODULE - II Coordinate Geometry CIRCLES. Notice the path in which the tip of the hand of a watch moves. (see Fig. 11.1)

Circles MODULE - II Coordinate Geometry CIRCLES. Notice the path in which the tip of the hand of a watch moves. (see Fig. 11.1) CIRCLES Notice the path in which the tip of the hand of a watch moves. (see Fig..) 0 9 3 8 4 7 6 5 Fig.. Fig.. Again, notice the curve traced out when a nail is fied at a point and a thread of certain

More information

Vector Supplement Part 1: Vectors

Vector Supplement Part 1: Vectors Vector Supplement Part 1: Vectors A vector is a quantity that has both magnitude and direction. Vectors are drawn as directed line segments and are denoted by boldface letters a or by a. The magnitude

More information

Solutions to Math 51 First Exam October 13, 2015

Solutions to Math 51 First Exam October 13, 2015 Solutions to Math First Exam October 3, 2. (8 points) (a) Find an equation for the plane in R 3 that contains both the x-axis and the point (,, 2). The equation should be of the form ax + by + cz = d.

More information

*X100/301* X100/301 MATHEMATICS HIGHER. Units 1, 2 and 3 Paper 1 (Non-calculator) Read Carefully

*X100/301* X100/301 MATHEMATICS HIGHER. Units 1, 2 and 3 Paper 1 (Non-calculator) Read Carefully X00/0 NATINAL QUALIFICATINS 007 TUESDAY, 5 MAY 9.00 AM 0.0 AM MATHEMATICS HIGHER Units, and Paper (Non-calculator) Read Carefull Calculators ma NT be used in this paper. Full credit will be given onl where

More information

MATH 1020 WORKSHEET 12.1 & 12.2 Vectors in the Plane

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

More information

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

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

More information

Chapter 13: Vectors and the Geometry of Space

Chapter 13: Vectors and the Geometry of Space Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic

More information

Chapter 13: Vectors and the Geometry of Space

Chapter 13: Vectors and the Geometry of Space Chapter 13: Vectors and the Geometry of Space 13.1 3-Dimensional Coordinate System 13.2 Vectors 13.3 The Dot Product 13.4 The Cross Product 13.5 Equations of Lines and Planes 13.6 Cylinders and Quadratic

More information

Section 1.5 Formal definitions of limits

Section 1.5 Formal definitions of limits Section.5 Formal definitions of limits (3/908) Overview: The definitions of the various tpes of limits in previous sections involve phrases such as arbitraril close, sufficientl close, arbitraril large,

More information

CHAPTER 3 : VECTORS. Definition 3.1 A vector is a quantity that has both magnitude and direction.

CHAPTER 3 : VECTORS. Definition 3.1 A vector is a quantity that has both magnitude and direction. EQT 101-Engineering Mathematics I Teaching Module CHAPTER 3 : VECTORS 3.1 Introduction Definition 3.1 A ector is a quantity that has both magnitude and direction. A ector is often represented by an arrow

More information

MAT389 Fall 2016, Problem Set 2

MAT389 Fall 2016, Problem Set 2 MAT389 Fall 2016, Problem Set 2 Circles in the Riemann sphere Recall that the Riemann sphere is defined as the set Let P be the plane defined b Σ = { (a, b, c) R 3 a 2 + b 2 + c 2 = 1 } P = { (a, b, c)

More information

Cumulative Review of Vectors

Cumulative Review of Vectors Cumulative Review of Vectors 1. For the vectors a! 1, 1, and b! 1, 4, 1, determine the following: a. the angle between the two vectors! the scalar and vector projections of a! on the scalar and vector

More information

Systems of Linear Equations

Systems of Linear Equations Systems of Linear Equations Linear Algebra MATH 2076 Linear Algebra SLEs Chapter 1 Section 1 1 / 8 Linear Equations and their Solutions A linear equation in unknowns (the variables) x 1, x 2,..., x n has

More information

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

Euclidean Spaces. Euclidean Spaces. Chapter 10 -S&B Chapter 10 -S&B The Real Line: every real number is represented by exactly one point on the line. The plane (i.e., consumption bundles): Pairs of numbers have a geometric representation Cartesian plane

More information

15. Eigenvalues, Eigenvectors

15. Eigenvalues, Eigenvectors 5 Eigenvalues, Eigenvectors Matri of a Linear Transformation Consider a linear ( transformation ) L : a b R 2 R 2 Suppose we know that L and L Then c d because of linearit, we can determine what L does

More information

CALCULUS 3 February 6, st TEST

CALCULUS 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 information

Exam 1 Review SOLUTIONS

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

More information

Math 261 Lecture Notes: Sections 6.1, 6.2, 6.3 and 6.4 Orthogonal Sets and Projections

Math 261 Lecture Notes: Sections 6.1, 6.2, 6.3 and 6.4 Orthogonal Sets and Projections Math 6 Lecture Notes: Sections 6., 6., 6. and 6. Orthogonal Sets and Projections We will not cover general inner product spaces. We will, however, focus on a particular inner product space the inner product

More information

1.2 LECTURE 2. Scalar Product

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

More information

9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b

9.1 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 information

vand v 3. Find the area of a parallelogram that has the given vectors as adjacent sides.

vand 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 information

VECTORS IN THREE DIMENSIONS

VECTORS IN THREE DIMENSIONS 1 CHAPTER 2. BASIC TRIGONOMETRY 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW VECTORS IN THREE DIMENSIONS 1 Vectors in Two Dimensions A vector is an object which has magnitude

More information

Name: SOLUTIONS Date: 09/07/2017. M20550 Calculus III Tutorial Worksheet 2

Name: SOLUTIONS Date: 09/07/2017. M20550 Calculus III Tutorial Worksheet 2 M20550 Calculus III Tutorial Worksheet 2 1. Find an equation of the plane passes through the point (1, 1, 7) and perpendicular to the line x = 1 + 4t, y = 1 t, z = 3. Solution: To write an equation of

More information

10.4 The Cross Product

10.4 The Cross Product Math 172 Chapter 10B notes Page 1 of 9 10.4 The Cross Product The cross product, or vector product, is defined in 3 dimensions only. Let aa = aa 1, aa 2, aa 3 bb = bb 1, bb 2, bb 3 then aa bb = aa 2 bb

More information

( 1 ) Find the co-ordinates of the focus, length of the latus-rectum and equation of the directrix of the parabola x 2 = - 8y.

( 1 ) Find the co-ordinates of the focus, length of the latus-rectum and equation of the directrix of the parabola x 2 = - 8y. PROBLEMS 04 - PARABOLA Page 1 ( 1 ) Find the co-ordinates of the focus, length of the latus-rectum and equation of the directrix of the parabola x - 8. [ Ans: ( 0, - ), 8, ] ( ) If the line 3x 4 k 0 is

More information

Vector Algebra August 2013

Vector Algebra August 2013 Vector Algebra 12.1 12.2 28 August 2013 What is a Vector? A vector (denoted or v) is a mathematical object possessing both: direction and magnitude also called length (denoted ). Vectors are often represented

More information

4.1 Distance and Length

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

More information

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

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

More information

Math 261 Solutions To Sample Exam 2 Problems

Math 261 Solutions To Sample Exam 2 Problems Solutions to Sample Eam Problems Math 6 Math 6 Solutions To Sample Eam Problems. Given to the right is the graph of a portion of four curves:,, and + 4. Note that these curves divide the plane into separate

More information

REVIEW 2, MATH 3020 AND MATH 3030

REVIEW 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 information

Math 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, 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 information

the Further Mathematics network

the Further Mathematics network the Further Mathematics network www.fmnetwork.org.uk 1 the Further Mathematics network www.fmnetwork.org.uk Further Pure 3: Teaching Vector Geometry Let Maths take you Further 2 Overview Scalar and vector

More information

Culminating Review for Vectors

Culminating Review for Vectors Culminating Review for Vectors 0011 0010 1010 1101 0001 0100 1011 An Introduction to Vectors Applications of Vectors Equations of Lines and Planes 4 12 Relationships between Points, Lines and Planes An

More information

LINEAR LAW. 1.1 Draw the line of best fit 1 2. x 2. log 10 y. y 2. log 10 x. 1.2 Write the equation for the line of best fit of the following graphs.

LINEAR LAW. 1.1 Draw the line of best fit 1 2. x 2. log 10 y. y 2. log 10 x. 1.2 Write the equation for the line of best fit of the following graphs. LINEAR LAW. Draw the line of best fit 3 4 log log. Write the equation for the line of best fit of the following graphs.. P(,3). Q(6,3).. P(,5) Q(,). [ 5 3 3 [ 5 5 Linear law 3. P(-,4). Q(5,6) 4. P(,8).

More information

VECTORS IN COMPONENT FORM

VECTORS 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

Worksheet #1. A little review.

Worksheet #1. A little review. Worksheet #1. A little review. I. Set up BUT DO NOT EVALUATE definite integrals for each of the following. 1. The area between the curves = 1 and = 3. Solution. The first thing we should ask ourselves

More information

Department of Physics IIT Kanpur, Semester II,

Department of Physics IIT Kanpur, Semester II, Department of Phsics IIT Kanpur, Semester II, 7-8 PHYA: Phsics II Solutions # Instructors: AKJ & SC Solution.: (a) At the top of the hill, the gradient of the height function should be ero, that is, h(,

More information

The Cross Product. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan The Cross Product

The Cross Product. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan The Cross Product The Cross Product MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Introduction Recall: the dot product of two vectors is a scalar. There is another binary operation on vectors

More information

12 VECTOR GEOMETRY Vectors in the Plane

12 VECTOR GEOMETRY Vectors in the Plane 1 VECTOR GEOMETRY 1.1 Vectors in the Plane Preliminar Questions 1. Answer true or false. Ever nonero vector is: (a) equivalent to a vector based at the origin. (b) equivalent to a unit vector based at

More information

Announcements. Topics: Homework: - section 12.4 (cross/vector product) - section 12.5 (equations of lines and planes)

Announcements. Topics: Homework: - section 12.4 (cross/vector product) - section 12.5 (equations of lines and planes) Topics: Announcements - section 12.4 (cross/vector product) - section 12.5 (equations of lines and planes) Homework: ü review lecture notes thoroughl ü work on eercises from the tetbook in sections 12.4

More information

Some linear transformations on R 2 Math 130 Linear Algebra D Joyce, Fall 2013

Some linear transformations on R 2 Math 130 Linear Algebra D Joyce, Fall 2013 Some linear transformations on R 2 Math 3 Linear Algebra D Joce, Fall 23 Let s look at some some linear transformations on the plane R 2. We ll look at several kinds of operators on R 2 including reflections,

More information

Vectors are used to represent quantities such as force and velocity which have both. and. The magnitude of a vector corresponds to its.

Vectors 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 information