Section 9.5: Equations of Lines and Planes


 Nickolas Parsons
 2 years ago
 Views:
Transcription
1 Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 35 odd, 237 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that is parallel to the vector v = < a, b, c > z L Q = ( x, y, z) r =< x, y, z > P = ( x, y, ) r =< x, y >, v = < a, b, c > y x The line L consists of all points Q = (x, y, z) for which the vector PQ is parallel to v. Now, PQ =< x x, y y z z, > Since PQ is parallel to v = < a, b, c >, PQ = t v where t is a scalar. Thus
2 2 Rewriting this equation gives < x, y y z z >= PQ = t v = < t a, t b, t c > x, < x y, z > < x, y, z >= t < a, b, c >, Solving for the vector < x, y, z > gives < x y, z >=< x, y, z > + t < a, b, c >, Setting r = < x, y, z >, r = < x, y, >, and v = < a, b, c >, we get the following vector equation of a line. Vector Equation of a Line in 3D Space The vector equation of a line in 3D space is given by the equation r = r + t v where r = < x, y, > is a vector whose components are made of the point ( x, y, ) on the line L and v = < a, b, c > are components of a vector that is parallel to the line L. If we take the vector equation < x y, z >=< x, y, z > + t < a, b, c >, and rewrite the right hand side of this equation as one vector, we obtain <, y, z >=< x + ta, y + tb z + tc > x, Equating components of this vector gives the parametric equations of a line.
3 3 Parametric Equations of a Line in 3D Space The parametric equations of a line L in 3D space are given by x x + ta, y = y + tb, z = z + tc, = where ( x, y, ) is a point passing through the line and v = < a, b, c > is a vector that the line is parallel to. The vector v = < a, b, c > is called the direction vector for the line L and its components a, b, and c are called the direction numbers. Assuming a, b, c, if we take each parametric equation and solve for the variable t, we obtain the equations t = x x a, t = y y b, t = z z c Equating each of these equations gives the symmetric equations of a line. Symmetric Equations of a Line in 3D Space The symmetric equations of a line L in 3D space are given by x x y y z = = a b c where ( x, y, ) is a point passing through the line and v = < a, b, c > is a vector that the line is parallel to. The vector v = < a, b, c > is called the direction vector for the line L and its components a, b, and c are called the direction numbers. Note!! To write the equation of a line in 3D space, we need a point on the line and a parallel vector to the line.
4 Example : Find the vector, parametric, and symmetric equations for the line through the point (,, 3) and parallel to the vector 2 i  4 j + 5 k. 4
5 5 Example 2: Find the parametric and symmetric equations of the line through the points (, 2, ) and (5, 4, 2) Solution: To find the equation of a line in 3D space, we must have at least one point on the line and a parallel vector. We already have two points one line so we have at least one. To find a parallel vector, we can simplify just use the vector that passes between the two given points, which will also be on this line. That is, if we assign the point P = (, 2, ) and Q = (5, 4, 2), then the parallel vector v is given by v PQ = < 5, 4 2, 2 > = < 6, 2, 2 > = Recall that the parametric equations of a line are given by x x + ta, y = y + tb, z = z + tc. = We can use either point P or Q as our point on the line ( x, y, ). We choose the point P and assign ( x, y, ) = (,2, ). The terms a, b, and c are the components of our parallel vector given by v = < 6, 2, 2 > found above. Hence a = 6, b = 2, and c = 2. Thus, the parametric equation of our line is given by or x = + t ( 6), y = 2 + t (2), z = + t (2) x = 6t, y = 2 + 2t, z = 2t To find the symmetric equations, we solve each parametric equation for t. This gives x y 2 t =, t =, t = z Setting these equations equal gives the symmetric equations. x y 2 = = z The graph on the following page illustrates the line we have found
6 6 It is important to note that the equations of lines in 3D space are not unique. In Example 2, for instance, had we used the point Q = (5, 4, 2) to represent the equation of the line with the parallel vector v = < 6, 2, 2 >, the parametric equations becomes x = 5 6t, y = 4 + 2t, z = 2 + 2t
7 7 Example 3: Find the parametric and symmetric equations of the line passing through the point (3, 5, 4) and parallel to the line x = + 3t, y =  2t, z = 3 + t. Solution:
8 8 Planes in 3D Space Consider the plane containing the point P = ( x, y, ) and normal vector n = < a, b, c > perpendicular to the plane. z n = < a, b, c > P = ( x, y, ) Q = ( x, y, z) y x The plane consists of all points Q = (x, y, z) for which the vector normal vector n = < a, b, c >. Since hold: PQ is orthogonal to the PQ and n are orthogonal, the following equations n PQ = <. a. b. c > < x x, y y, z >= a ( x x ) + b( y y ) + c( z ) = This gives the standard equation of a plane. If we expand this equation we obtain the following equation: ax + by + cz ax by cz = Constant d
9 9 Setting d = ax by c gives the general form of the equation of a plane in 3D space ax + by + cz + d =. We summarize these results as follows. Standard and General Equations of a Plane in the 3D space The standard equation of a plane in 3D space has the form a ( x x ) + b( y y ) + c( z ) = where ( x, y, ) is a point on the plane and n = < a, b, c > is a vector normal (orthogonal to the plane). If this equation is expanded, we obtain the general equation of a plane of the form ax + by + cz + d = Note!! To write the equation of a plane in 3D space, we need a point on the plane and a vector normal (orthogonal) to the plane.
10 Example 4: Find the equation of the plane through the point (4, 3, ) that is perpendicular to the vector a = 4 i + 7 j 2 k. Solution:
11 Example 5: Find the equation of the plane passing through the points (, 2, 3), (2, 3, ), and (, 2, ). Solution:
12 2 Intersecting Planes Suppose we are given two intersecting planes with angle θ between them. n θ n 2 Plane θ Plane 2 Let n and n 2 be normal vectors to these planes. Then n n2 cosθ = n n 2 Thus, two planes are. Perpendicular if n n, which implies 2 = 2. Parallel if n 2 = cn, where c is a scalar. π θ =. 2 Notes. Given the general equation of a plane ax + by + cz + d =, the normal vector is n = < a, b, c >. 2. The intersection of two planes is a line.
13 3 Example 6: Determine whether the planes 3 x + y 4z = 3 and 9 x 3y + 2z = 4 are orthogonal, parallel, or neither. Find the angle of intersection and the set of parametric equations for the line of intersection of the plane. Solution:
14 4 Example 7: Determine whether the planes x 3 y + 6z = 4 and 5 x + y z = 4 are orthogonal, parallel, or neither. Find the angle of intersection and the set of parametric equations for the line of intersection of the plane. Solution: For the plane x 3 y + 6z = 4, the normal vector is n = <, 3, 6 > and for the plane 5 x + y z = 4, the normal vector is n 2 = < 5,, >. The two planes will be orthogonal only if their corresponding normal vectors are orthogonal, that is, if n n. However, we see that 2 = n n2 =<, 3,6 > < 5,, >= ()(5) + ( 3)() + (6)( ) = = 4 Hence, the planes are not orthogonal. If the planes are parallel, then their corresponding normal vectors must be parallel. For that to occur, there must exist a scalar k where n 2 = k n Rearranging this equation as k n = n 2 and substituting for n and n 2 gives or k <, 3,6 > = < 5,, > < k, 3k,6k > = < 5,, >. Equating components gives the equations k = 5,  3k =, 6k =  which gives k = 5, k =, k = Since the values of k are not the same for each component to make the vector n 2 a scalar multiple of the vector n, the planes are not parallel. Thus, the planes must intersect in a straight line at a given angle. To find this angle, we use the equation For this formula, we have the following: n n2 cosθ = n n n n2 =<, 3,6 > < 5,, >= ()(5) + ( 3)() + (6)( ) = = n = () + ( 3) + (6) = = = 2 n = (5) + () + ( ) = (continued on next page) 46 2
15 5 Thus, Solving for θ gives cosθ = θ = cos.68 radians To find the equation of the line of intersection between the two planes, we need a point on the line and a parallel vector. To find a point on the line, we can consider the case where the line touches the xy plane, that is, where z =. If we take the two equations of the plane x 3 y + 6z = 4 5 x + y z = 4 and substitute z =, we obtain the system of equations Taking the first equation and multiplying by 5 gives x 3 y = 4 () 5 x + y = 4 (2) 5x + 5y = 2 5 x + y = 4 6 Adding the two equations gives 6y = 6 or y = =. Substituting y = back 6 into equation () gives x 3 ( ) = 4 or x + 3 = 4. Solving for x gives x = 43 =. Thus, the point on the plane is (, , ). To find a parallel vector for the line, we use the fact that since the line is on both planes, it must be orthogonal to both normal vectors n and n 2. Since the cross product n n2 gives a vector orthogonal to both n and n 2, n will be a parallel vector for the line. Thus, we say that n 2 v = n n 2 = i 5 j 3 k 6 = i 3 6 j k 53 = i (3 6) = 3i + 3j + 6k j( 3) + k( 5) (continued on next page)
16 6 Hence, using the point (, , ) and the parallel vector parametric equations of the line are v = 3 i + 3j + 6k, we find the x = 3t, y = + 3 t, z = 6t The following shows a graph of the two planes and the line we have found. Example 8: Find the point where the line x = + t, y = 2t, and z = 3t intersects the plane 4x + 2y 4z = 2. Solution:
17 7 Distance Between Points and a Plane Suppose we are given a point Q not in a plane and a point P on the plane and our goal is to find the shortest distance between the point Q and the plane. n Q P By projecting the vector comp n PQ ), we can find the distance D. PQ onto the normal vector n (calculating the scalar projection Distance Between Q = D = and the plane comp n PQ n PQ = n
18 8 Example 9: Find the distance between the point (, 2, 3) and line 2 x y + z = 4. Solution: Since we are given the point Q = (, 2, 3), we need to find a point on the plane 2 x y + z = 4 in order to find the vector PQ. We can simply do this by setting y = and z = in the plane equation and solving for x. Thus we have Thus P = (2,, ) and the vector PQ is 2 x y + z = 4 2 x + = 4 2x = 4 x = 4 2 = 2 PQ =< 2,2,3 >=<,2,3 >. Hence, using the fact that the normal vector for the plane is n =< 2,, >, we have Distance Between PQ n <,2,3 > < 2,, > = = Q and the plane n (2) + ( ) + () = = = Thus, the distance is. 6
Equations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More informationLINES AND PLANES CHRIS JOHNSON
LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3space, as well as define the angle between two nonparallel planes, and determine the distance
More informationSection 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables
The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,
More informationSection 13.5 Equations of Lines and Planes
Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines  specifically, tangent lines.
More information12.5 Equations of Lines and Planes
Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P
More information= y y 0. = z z 0. (a) Find a parametric vector equation for L. (b) Find parametric (scalar) equations for L.
Math 21a Lines and lanes Spring, 2009 Lines in Space How can we express the equation(s) of a line through a point (x 0 ; y 0 ; z 0 ) and parallel to the vector u ha; b; ci? Many ways: as parametric (scalar)
More information10.5. Click here for answers. Click here for solutions. EQUATIONS OF LINES AND PLANES. 3x 4y 6z 9 4, 2, 5. x y z. z 2. x 2. y 1.
SECTION EQUATIONS OF LINES AND PLANES 1 EQUATIONS OF LINES AND PLANES A Click here for answers. S Click here for solutions. 1 Find a vector equation and parametric equations for the line passing through
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.114.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two nonzero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More informationJUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson
JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More informationDot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product
Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level  MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: 97 Mathematics
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationMathematics 205 HWK 6 Solutions Section 13.3 p627. Note: Remember that boldface is being used here, rather than overhead arrows, to indicate vectors.
Mathematics 205 HWK 6 Solutions Section 13.3 p627 Note: Remember that boldface is being used here, rather than overhead arrows, to indicate vectors. Problem 5, 13.3, p627. Given a = 2j + k or a = (0,2,
More informationL 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
More informationSection 2.4: Equations of Lines and Planes
Section.4: Equations of Lines and Planes An equation of three variable F (x, y, z) 0 is called an equation of a surface S if For instance, (x 1, y 1, z 1 ) S if and only if F (x 1, y 1, z 1 ) 0. x + y
More informationLINES AND PLANES IN R 3
LINES AND PLANES IN R 3 In this handout we will summarize the properties of the dot product and cross product and use them to present arious descriptions of lines and planes in three dimensional space.
More informationCIRCLE COORDINATE GEOMETRY
CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle
More informationMAT 1341: REVIEW II SANGHOON BAEK
MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and
More information1.5 Equations of Lines and Planes in 3D
40 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3D Recall that given a point P = (a, b, c), one can draw a vector from
More informationSection 8.8. 1. The given line has equations. x = 3 + t(13 3) = 3 + 10t, y = 2 + t(3 + 2) = 2 + 5t, z = 7 + t( 8 7) = 7 15t.
. The given line has equations Section 8.8 x + t( ) + 0t, y + t( + ) + t, z 7 + t( 8 7) 7 t. The line meets the plane y 0 in the point (x, 0, z), where 0 + t, or t /. The corresponding values for x and
More informationSection 11.4: Equations of Lines and Planes
Section 11.4: Equations of Lines and Planes Definition: The line containing the point ( 0, 0, 0 ) and parallel to the vector v = A, B, C has parametric equations = 0 + At, = 0 + Bt, = 0 + Ct, where t R
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More information13.4 THE CROSS PRODUCT
710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product
More information(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,
Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We
More informationMath 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t
Math 241 Lines and Planes (Solutions) The equations for planes P 1, P 2 and P are P 1 : x 2y + z = 7 P 2 : x 4y + 5z = 6 P : (x 5) 2(y 6) + (z 7) = 0 The equations for lines L 1, L 2, L, L 4 and L 5 are
More informationChapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis
Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >
More informationLines and Planes in R 3
.3 Lines and Planes in R 3 P. Daniger Lines in R 3 We wish to represent lines in R 3. Note that a line may be described in two different ways: By specifying two points on the line. By specifying one point
More informationDetermine whether the following lines intersect, are parallel, or skew. L 1 : x = 6t y = 1 + 9t z = 3t. x = 1 + 2s y = 4 3s z = s
Homework Solutions 5/20 10.5.17 Determine whether the following lines intersect, are parallel, or skew. L 1 : L 2 : x = 6t y = 1 + 9t z = 3t x = 1 + 2s y = 4 3s z = s A vector parallel to L 1 is 6, 9,
More informationCHAPTER FIVE. 5. Equations of Lines in R 3
118 CHAPTER FIVE 5. Equations of Lines in R 3 In this chapter it is going to be very important to distinguish clearly between points and vectors. Frequently in the past the distinction has only been a
More informationMath 215 HW #6 Solutions
Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T
More informationReview Sheet for Test 1
Review Sheet for Test 1 Math 26100 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And
More informationGeometric description of the cross product of the vectors u and v. The cross product of two vectors is a vector! u x v is perpendicular to u and v
12.4 Cross Product Geometric description of the cross product of the vectors u and v The cross product of two vectors is a vector! u x v is perpendicular to u and v The length of u x v is uv u v sin The
More informationCross product and determinants (Sect. 12.4) Two main ways to introduce the cross product
Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Geometrical definition Properties Expression in components. Definition in components Properties Geometrical expression.
More informationSection 1.8 Coordinate Geometry
Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
More informationDefinition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.
6.1 Vectors in the Plane PreCalculus 6.1 VECTORS IN THE PLANE Learning Targets: 1. Find the component form and the magnitude of a vector.. Perform addition and scalar multiplication of two vectors. 3.
More information9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes
The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is socalled because when the scalar product of
More informationExam 1 Sample Question SOLUTIONS. y = 2x
Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More information2.1 Three Dimensional Curves and Surfaces
. Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two or threedimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The
More informationdiscuss how to describe points, lines and planes in 3 space.
Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position
More informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationSolutions for Review Problems
olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector
More informationEquations of Lines and Planes
Calculus 3 Lia Vas Equations of Lines and Planes Planes. A plane is uniquely determined by a point in it and a vector perpendicular to it. An equation of the plane passing the point (x 0, y 0, z 0 ) perpendicular
More informationJim Lambers MAT 169 Fall Semester 200910 Lecture 25 Notes
Jim Lambers MAT 169 Fall Semester 00910 Lecture 5 Notes These notes correspond to Section 10.5 in the text. Equations of Lines A line can be viewed, conceptually, as the set of all points in space that
More informationParametric Curves. EXAMPLE: Sketch and identify the curve defined by the parametric equations
Section 9. Parametric Curves 00 Kiryl Tsishchanka Parametric Curves Suppose that x and y are both given as functions of a third variable t (called a parameter) by the equations x = f(t), y = g(t) (called
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationTwo vectors are equal if they have the same length and direction. They do not
Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must
More informationVector Algebra CHAPTER 13. Ü13.1. Basic Concepts
CHAPTER 13 ector Algebra Ü13.1. Basic Concepts A vector in the plane or in space is an arrow: it is determined by its length, denoted and its direction. Two arrows represent the same vector if they have
More informationProblem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.
Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve
More informationMATH2210 Notebook 1 Fall Semester 2016/2017. 1 MATH2210 Notebook 1 3. 1.1 Solving Systems of Linear Equations... 3
MATH0 Notebook Fall Semester 06/07 prepared by Professor Jenny Baglivo c Copyright 009 07 by Jenny A. Baglivo. All Rights Reserved. Contents MATH0 Notebook 3. Solving Systems of Linear Equations........................
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationi=(1,0), j=(0,1) in R 2 i=(1,0,0), j=(0,1,0), k=(0,0,1) in R 3 e 1 =(1,0,..,0), e 2 =(0,1,,0),,e n =(0,0,,1) in R n.
Length, norm, magnitude of a vector v=(v 1,,v n ) is v = (v 12 +v 22 + +v n2 ) 1/2. Examples v=(1,1,,1) v =n 1/2. Unit vectors u=v/ v corresponds to directions. Standard unit vectors i=(1,0), j=(0,1) in
More informationGCE Mathematics (6360) Further Pure unit 4 (MFP4) Textbook
Version 36 klm GCE Mathematics (636) Further Pure unit 4 (MFP4) Textbook The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales 364473 and a
More information1.7 Cylindrical and Spherical Coordinates
56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a twodimensional coordinate system in which the
More information1.(6pts) Find symmetric equations of the line L passing through the point (2, 5, 1) and perpendicular to the plane x + 3y z = 9.
.(6pts Find symmetric equations of the line L passing through the point (, 5, and perpendicular to the plane x + 3y z = 9. (a x = y + 5 3 = z (b x (c (x = ( 5(y 3 = z + (d x (e (x + 3(y 3 (z = 9 = y 3
More information... ... . (2,4,5).. ...
12 Three Dimensions ½¾º½ Ì ÓÓÖ Ò Ø ËÝ Ø Ñ So far wehave been investigatingfunctions ofthe form y = f(x), withone independent and one dependent variable Such functions can be represented in two dimensions,
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationName Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155
Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate
More informationC relative to O being abc,, respectively, then b a c.
2 EPProgram  Strisuksa School  Roiet Math : Vectors Dr.Wattana Toutip  Department of Mathematics Khon Kaen University 200 :Wattana Toutip wattou@kku.ac.th http://home.kku.ac.th/wattou 2. Vectors A
More informationSection 2.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates
Section.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates The polar coordinate system consists of the origin O,the rotating ray or half line from O with unit tick. A point P in
More informationMA261A Calculus III 2006 Fall Homework 3 Solutions Due 9/22/2006 8:00AM
MA6A Calculus III 6 Fall Homework Solutions Due 9//6 :AM 9. # Find the parametric euation and smmetric euation for the line of intersection of the planes + + z = and + z =. To write down a line euation,
More information( 1)2 + 2 2 + 2 2 = 9 = 3 We would like to make the length 6. The only vectors in the same direction as v are those
1.(6pts) Which of the following vectors has the same direction as v 1,, but has length 6? (a), 4, 4 (b),, (c) 4,, 4 (d), 4, 4 (e) 0, 6, 0 The length of v is given by ( 1) + + 9 3 We would like to make
More information[1] Diagonal factorization
8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:
More information5 VECTOR GEOMETRY. 5.0 Introduction. Objectives. Activity 1
5 VECTOR GEOMETRY Chapter 5 Vector Geometry Objectives After studying this chapter you should be able to find and use the vector equation of a straight line; be able to find the equation of a plane in
More informationMathematics Notes for Class 12 chapter 10. Vector Algebra
1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is nonnegative
More informationMATH 275: Calculus III. Lecture Notes by Angel V. Kumchev
MATH 275: Calculus III Lecture Notes by Angel V. Kumchev Contents Preface.............................................. iii Lecture 1. ThreeDimensional Coordinate Systems..................... 1 Lecture
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationa.) Write the line 2x  4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a
Bellwork a.) Write the line 2x  4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a c.) Find the slope of the line perpendicular to part b or a May 8 7:30 AM 1 Day 1 I.
More informationis in plane V. However, it may be more convenient to introduce a plane coordinate system in V.
.4 COORDINATES EXAMPLE Let V be the plane in R with equation x +2x 2 +x 0, a twodimensional subspace of R. We can describe a vector in this plane by its spatial (D)coordinates; for example, vector x 5
More informationTangent line of a circle can be determined once the tangent point or the slope of the line is known.
Worksheet 7: Tangent Line of a Circle Name: Date: Tangent line of a circle can be determined once the tangent point or the slope of the line is known. Straight line: an overview General form : Ax + By
More informationProblem set on Cross Product
1 Calculate the vector product of a and b given that a= 2i + j + k and b = i j k (Ans 3 j  3 k ) 2 Calculate the vector product of i  j and i + j (Ans ) 3 Find the unit vectors that are perpendicular
More informationwww.sakshieducation.com
LENGTH OF THE PERPENDICULAR FROM A POINT TO A STRAIGHT LINE AND DISTANCE BETWEEN TWO PAPALLEL LINES THEOREM The perpendicular distance from a point P(x 1, y 1 ) to the line ax + by + c 0 is ax1+ by1+ c
More information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the ndimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More informationSolutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
More informationExample SECTION 131. XAXIS  the horizontal number line. YAXIS  the vertical number line ORIGIN  the point where the xaxis and yaxis cross
CHAPTER 13 SECTION 131 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants XAXIS  the horizontal
More informationUnit 5: Coordinate Geometry Practice Test
Unit 5: Coordinate Geometry Practice Test Math 10 Common Name: Block: Please initial this box to indicate you carefully read over your test and checked your work for simple mistakes. What I can do in this
More informationGeometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi
Geometry of Vectors Carlo Tomasi This note explores the geometric meaning of norm, inner product, orthogonality, and projection for vectors. For vectors in threedimensional space, we also examine the
More information1. Equations for lines on the plane and planes in the space.
1. Equations for lines on the plane and planes in the space. 1.1. General implicit vector equation. (1) a r=α This equation defines a line in the plane and a plane in the 3space. Here r is the radiusvector
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals,
More informationSection 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 2537, 40, 42, 44, 45, 47, 50
Section 11.1: Vectors in the Plane Page 779 Suggested Problems: 1, 5, 9, 17, 3, 537, 40, 4, 44, 45, 47, 50 Determine whether the following vectors a and b are perpendicular. 5) a = 6, 0, b = 0, 7 Recall
More informationVectors, Gradient, Divergence and Curl.
Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationThe PointSlope Form
7. The PointSlope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope
More information42 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. Figure 1.18: Parabola y = 2x 2. 1.6.1 Brief review of Conic Sections
2 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.18: Parabola y = 2 1.6 Quadric Surfaces 1.6.1 Brief review of Conic Sections You may need to review conic sections for this to make more sense. You
More informationLecture 9: Lines. m = y 2 y 1 x 2 x 1
Lecture 9: Lines If we have two distinct points in the Cartesian plane, there is a unique line which passes through the two points. We can construct it by joining the points with a straight edge and extending
More information*X100/12/02* X100/12/02. MATHEMATICS HIGHER Paper 1 (Noncalculator) MONDAY, 21 MAY 1.00 PM 2.30 PM NATIONAL QUALIFICATIONS 2012
X00//0 NTIONL QULIFITIONS 0 MONY, MY.00 PM.0 PM MTHEMTIS HIGHER Paper (Noncalculator) Read carefully alculators may NOT be used in this paper. Section Questions 0 (40 marks) Instructions for completion
More informationVector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.
1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called
More informationThe Dot and Cross Products
The Dot and Cross Products Two common operations involving vectors are the dot product and the cross product. Let two vectors =,, and =,, be given. The Dot Product The dot product of and is written and
More informationLesson 19: Equations for Tangent Lines to Circles
Student Outcomes Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line
More information