LINES AND PLANES CHRIS JOHNSON


 Catherine Davis
 3 years ago
 Views:
Transcription
1 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 from a point to a plane using properties of vector projections. 1. Lines 1.1. Vector, Parametric, and Symmetric Equations. Recall that in two dimensions, to specify a line you need two pieces of information: a point the line passes through, and the slope of the line. The slope of the line really just tells us the direction the line points in. In three dimensions we also need two pieces of information to determine a line; if we want to give the equation of a line, then we need to know a point on the line and the direction of the line. While in two dimensions we could use a single number (the slope) to determine the direction of the line, in three dimensions we ll use vectors. (Really, the slope in two dimensions determines a vector: a slope of m is the same as a vector 1, m. Since the first component of this vector is always 1, it s only the second component that matters.) So let s say, to keep things easy, we want the equation of a line through the origin. Let s suppose the vector v = a, b, c points along the line. This means that if a point P = (x, y, z) lives on the line, we could perform the scalar multiplication t v to stretch v out enough so that its tip was at P. That is, the displacement vector r = OP = x, y, z is related to the direction vector v = a, b, c by the following equation: r = t v This is the vector equation of the line through the origin in direction v. Now let s suppose we want our line to pass through some point P 0 = (x 0, y 0, z 0 ) instead of the origin. Let r 0 denote the displacement vector r 0 = OP 0 = x 0, y 0, z 0. All we need to do is take our line through the origin, and translate it along the vector r 0 so that the line passes through P 0. So, if another point P = (x, y, z) is to be on the line, 1
2 2 CHRIS JOHNSON letting r denote the vector x, y, z, we have the following equation: r = r 0 + t v. This is the vector equation of the line through P 0 in the direction of v. If we write this out in terms of components we have x, y, z = x 0, y 0, z 0 + ta, tb, tc = x 0 + ta, y 0 + ta, z 0 + ta. Two vectors are equal if and only if their components are equal, so by equating components we actually have three equations: x =x 0 + ta y =y 0 + tb z =z 0 + tc. Notice that on the righthand side the values a, b, c, x 0, y 0, z 0 are all fixed: they don t change once we say what the point P 0 is and where the vector v points. The t is the only thing on the righthand side that can change. Thus the values x, y, z on the lefthand side are functions of t. So really the above equations should be written as x(t) =x 0 + ta y(t) =y 0 + tb z(t) =z 0 + tc. These three equations form the parametric equations of the line through P 0 = (x 0, y 0, z 0 ) in the direction of v = a, b, c. Example 1.1. Find the vector and parametric equations for the line through the point (3, 4, 1) in the direction of 1 /2, 1, 3. The vector equation is x, y, z = 3 + t2, 4 t, 1 + 3t. To get the parametric equations, just equate the components to get: x(t) =3 + t 2 y(t) = 4 t z(t) =1 + 3t.
3 LINES AND PLANES 3 Now, assuming none of a, b, or c is zero, we could solve each of our equations above for t to get t = x x 0 a t = y y 0 b t = z z 0. c Since each of the things on the righthand side equals t, these quantities are all equal. Thus we have x x 0 a = y y 0 b = z z 0. c These are the symmetric equations of the line through P 0 = (x 0, y 0, z 0 ) in the direction of v = a, b, c. Again, notice these equations only make sense if none of a, b, or c equals zero. (If one of them did equal zero, we d have division by zero.) In our example above, the symmetric equations are x 3 1/2 = y = z 1 3. These symmetric equations are nice because they give us an easy way to determine if a point is on the line or not. Example 1.2. Are the points (4, 6, 7) and ( 5, 0, 14) on the line through (3, 4, 1) in the direction of 1 /2, 1, 3? This is our line from the last example, so the symmetric equations are given above. We plug in the coordinates of each point to see if we have equality or not. In the case of (4, 2, 7) we have: /2 = = Each of these expressions is 2, so we have equality, and the point (4, 2, 7) is on the line. Plugging in ( 5, 0, 14) we have /2 = The first two expressions equal 4, while the last one equals 5. Thus the point ( 5, 0, 14) is not on the line.
4 4 CHRIS JOHNSON 1.2. Parallel and Skew Lines. Just as in two dimensions, two lines are parallel if they point in the same direction. In the case of two dimensions this meant that the two lines had the same slope. In the case of three dimensions it means that the two direction vectors are scalar multiples of one another. For example, the lines x 2 3 = y = z 1 2 and x = y 3 = z 2 6 are parallel. These lines have respective direction vectors 3, 1, 2 and 9, 3, 6. The second one is 2 times the first. In the case of two dimensions, two lines are parallel if and only if they never touch. This is not the case in three dimensions. Consider the lines with the following parametric equations. x 1 (t) =3 t y 1 (t) =1 + 2t z 1 (t) = 1 x 1 (t) =4 t y 1 (t) =2 + t z 1 (t) =1 Let s call these lines L 1 and L 2. From the equations we can already tell that the first line passes through the point (3, 1, 1) in the direction of v 1 = 1, 2, 0, and the second line passes through the point (4, 2, 1) in the direction of v 2 = 1, 1, 0. Notice that L 1 is contained in the plane z = 1, and L 2 is in the plane z = 1. Thus there s no possible way these lines can ever intersect. However, their direction vectors are not scalar multiples of one another: there is no λ R that makes v 1 = λ v 2. So these lines are not parallel, nor do they intersect. When this happens we say that the lines are skew. 2. Planes 2.1. Equations of Planes. Now we consider planes. Of course we ve already seen several examples of equations of planes in previous lectures, but everything we looked at before was a very special case (e.g., the coordinate planes). We want to be able to determine the equation
5
6 6 CHRIS JOHNSON where d = (n 1 x 0 + n 2 y 0 + n 3 z 0 ). This is a linear equation, and just like the symmetric equations for lines, they give us a really easy way to determine if a point is on a plane or not. Example 2.1. Find the linear equation of the plane which contains the point ( 2, 3, 5) and whose normal vector is 7, 2, 3. Is the point (1, 3, 4) on this plane? To get the linear equation: 7(x + 2) + 2(y 3) + 3(z + 5) = 0 = 7x y 6 + 3z + 15 = 0 = 7x + 2y + 3z + 23 = 0 Now we check if (1, 3, 4) is on the plane or not: So the point is not on the plane ( 4) + 23 = Distance to a Plane. Recall that if P = (x, y, z) is some point in 3space, we can measure the distance from that point to each of the coordinate planes by first projecting onto the plane, and then measuring the distance from the projection to our initial point. We can do the exact same thing but for other planes. Let s say that we re given a point P 1 = (x 1, y 1, z 1 ) and we want to measure the distance from this point to the plane ax + by + cz = d. What we want to do is find the point P 0 = (x 0, y 0, z 0 ) on the plane so that the displacement vector P 0 P 1 sticks orhotogonally out of the plane. That is, P 0 P 1 = λ n where n = a, b, c is the normal vector of the plane. We then need to figure out how long this displacement vector is. Let s suppose that P = (x, y, z) is any other point in this plane. Then if r = OP and r 0 = OP 0, we know n ( r r 0 ) = 0. Let s = P P 1. What we want to do is project s onto n, and measure the length of that vector that is, we want the absolute value of comp n s. In components, s = x 1 x, y 1 y, z 1 z. Thus the distance from
7 LINES AND PLANES 7 P 1 = (x 1, y 1, z 1 ) to the plane ax + by + cz = d is given by distance = comp n s = s n n = x 1 x, y 1 y, z 1 z a, b, c a2 + b 2 + c 2 = a(x 1 x) + b(y 1 y) + c(z 1 z) a2 + b 2 + c 2 = ax 1 + by 1 + cz 1 (ax + by + cz) a2 + b 2 + c 2 Now, since our point P = (x, y, z) lives on the plane, it satisfies the equation ax + by + cz = d, so the above becomes distance = ax 1 + by 1 + cz 1 d a2 + b 2 + c 2. Caution: The formula that appears in your book is slightly different than what s written here because your book assumes the equation of the plane is written as ax+by+cz+d = 0, which becomes ax+by+cz = d: our d is the negative of the d in the book. So you have to be a slightly cautious when using these formulas. Example 2.2. Find the distance from the point (1, 2, 4) to the plane 3x y + 5z = 6. Here the normal vector to plane is n = 3, 1, 5, d = 6, and (x 1, y 1, z 1 ) = (1, 2, 4). Plugging these values into our formula above, distance = ax 1 + by 1 + cz 1 (ax + by + cz) a2 + b 2 + c = = The Angle Between Two Planes. We ll say that two planes are parallel if their normal vectors are scalar multiples of one another.
8 8 CHRIS JOHNSON Example 2.3. Are the two planes below parallel? 3x + 6y 7z =13 x 2y z =0 From the equations, we can easily pull of the normal vectors: 3, 6, 7 and 1, 2, 7 /3. These vectors are obviously scalar multiples: Thus the planes are parallel. 3, 6, 7 = 3 1, 2, 7 /3. Example 2.4. Are the two planes below parallel? 2x y + z =3 4x y + 2x =6 Again, we pull off the normal vectors: 2, 1, 1 and 4, 1, 2. Notice these vectors can t be scalar multiples of one another: we d need to multiply the first vector by 2 to get a 4 in the first component of the second vector, but then this would give a 2 in the second component. Hence the planes are not parallel. If two planes are not parallel, we can talk about the angle at which those planes meet. Of course, geometrically, this is exactly what you think it should be, but how do you go about calculating this angle? It s relatively clear that the normal vectors to the planes meet at the same angle as the planes themselves, and we can calculate the angle between the vectors by using dot products. That is, if our planes have normal vectors n = n 1, n 2, n 3 and m = m 1, m 2, m 3, then the angle between the planes is the same as the angle between these vectors, which we know is just ( ) m n θ = cos 1. m n Example 2.5. What s the angle between the planes given by the equations below? 2x + 3y z =0 3x + y + 4z =12 Our normal vectors are 2, 3, 1 and 3, 1, 4. Notice these vectors aren t scalar multiples of one another, so the planes aren t parallel and must intersect. The angle between the planes is the same as the angle
9 LINES AND PLANES 9 between these two vectors: ( ) 2, 3, 1 3, 1, 4 θ = cos 1 2, 3, 1 3, 1, 4 ( ) = cos ( ) 3 = cos
Section 9.5: Equations of Lines and Planes
Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 35 odd, 237 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
More informationEquations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationVector Spaces; the Space R n
Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which
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 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 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 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 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 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 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 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 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 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 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 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 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 informationLinear Algebra Notes
Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note
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 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 informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More informationNotice that v v w (4)( 15) ( 3)( 20) (0)(2) ( 2)( 15) (2)( 20) (5)(2)
The Cross Product When discussing the dot product, we showed how two vectors can be combined to get a number. Here, we shall see another way of combining vectors, this time resulting in a vector. This
More 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 informationCopyrighted Material. Chapter 1 DEGREE OF A CURVE
Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two
More informationMathematics 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 informationVector has a magnitude and a direction. Scalar has a magnitude
Vector has a magnitude and a direction Scalar has a magnitude Vector has a magnitude and a direction Scalar has a magnitude a brick on a table Vector has a magnitude and a direction Scalar has a magnitude
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 informationVectors Math 122 Calculus III D Joyce, Fall 2012
Vectors Math 122 Calculus III D Joyce, Fall 2012 Vectors in the plane R 2. A vector v can be interpreted as an arro in the plane R 2 ith a certain length and a certain direction. The same vector can be
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 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 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 informationLecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20
Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding
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 informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More 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 informationMath 018 Review Sheet v.3
Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1  Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.
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 informationSection 10.4 Vectors
Section 10.4 Vectors A vector is represented by using a ray, or arrow, that starts at an initial point and ends at a terminal point. Your textbook will always use a bold letter to indicate a vector (such
More informationApplied Linear Algebra
Applied Linear Algebra OTTO BRETSCHER http://www.prenhall.com/bretscher Chapter 7 Eigenvalues and Eigenvectors ChiaHui Chang Email: chia@csie.ncu.edu.tw National Central University, Taiwan 7.1 DYNAMICAL
More information521493S Computer Graphics. Exercise 2 & course schedule change
521493S Computer Graphics Exercise 2 & course schedule change Course Schedule Change Lecture from Wednesday 31th of March is moved to Tuesday 30th of March at 1618 in TS128 Question 2.1 Given two nonparallel,
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 informationv w is orthogonal to both v and w. the three vectors v, w and v w form a righthanded set of vectors.
3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with
More 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 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 informationSection V.4: Cross Product
Section V.4: Cross Product Definition The cross product of vectors A and B is written as A B. The result of the cross product A B is a third vector which is perpendicular to both A and B. (Because the
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 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 informationFigure 2.1: Center of mass of four points.
Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would
More informationNotes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
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 informationChapter 11 Equilibrium
11.1 The First Condition of Equilibrium The first condition of equilibrium deals with the forces that cause possible translations of a body. The simplest way to define the translational equilibrium of
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 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 informationLecture 2: Homogeneous Coordinates, Lines and Conics
Lecture 2: Homogeneous Coordinates, Lines and Conics 1 Homogeneous Coordinates In Lecture 1 we derived the camera equations λx = P X, (1) where x = (x 1, x 2, 1), X = (X 1, X 2, X 3, 1) and P is a 3 4
More informationVectors 2. The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996.
Vectors 2 The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996. Launch Mathematica. Type
More informationVectorsAlgebra and Geometry
Chapter Two VectorsAlgebra and Geometry 21 Vectors A directed line segment in space is a line segment together with a direction Thus the directed line segment from the point P to the point Q is different
More informationVector Math Computer Graphics Scott D. Anderson
Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about
More 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 and normal lines to conics
4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints
More informationThe Gradient and Level Sets
The Gradient and Level Sets. Let f(x, y) = x + y. (a) Find the gradient f. Solution. f(x, y) = x, y. (b) Pick your favorite positive number k, and let C be the curve f(x, y) = k. Draw the curve on the
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 V.3: Dot Product
Section V.3: Dot Product Introduction So far we have looked at operations on a single vector. There are a number of ways to combine two vectors. Vector addition and subtraction will not be covered here,
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the yaxis We observe that
More informationAdditional Examples of using the Elimination Method to Solve Systems of Equations
Additional Examples of using the Elimination Method to Solve Systems of Equations. Adjusting Coecients and Avoiding Fractions To use one equation to eliminate a variable, you multiply both sides of that
More information2 Topics in 3D Geometry
2 Topics in 3D Geometry In two dimensional space, we can graph curves and lines. In three dimensional space, there is so much extra space that we can graph planes and surfaces in addition to lines and
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationLecture 6. Weight. Tension. Normal Force. Static Friction. Cutnell+Johnson: 4.84.12, second half of section 4.7
Lecture 6 Weight Tension Normal Force Static Friction Cutnell+Johnson: 4.84.12, second half of section 4.7 In this lecture, I m going to discuss four different kinds of forces: weight, tension, the normal
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 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 informationi=(1,0), j=(0,1) in R 2 i=(1,0,0), j=(0,1,0), k=(0,0,1) in R 3 e 1 =(1,0,..,0), e 2 =(0,1,,0),,e n =(0,0,,1) in R n.
Length, norm, magnitude of a vector v=(v 1,,v n ) is v = (v 12 +v 22 + +v n2 ) 1/2. Examples v=(1,1,,1) v =n 1/2. Unit vectors u=v/ v corresponds to directions. Standard unit vectors i=(1,0), j=(0,1) in
More information1.7 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 informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More 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 informationAssignment 3. Solutions. Problems. February 22.
Assignment. Solutions. Problems. February.. Find a vector of magnitude in the direction opposite to the direction of v = i j k. The vector we are looking for is v v. We have Therefore, v = 4 + 4 + 4 =.
More informationPractice Problems for Midterm 1
Practice Problems for Midterm 1 Here are some problems for you to try. A few I made up, others I found from a variety of sources, including Stewart s Multivariable Calculus book. (1) A boy throws a football
More information