Equations Involving Lines and Planes Standard equations for lines in space

Size: px
Start display at page:

Download "Equations Involving Lines and Planes Standard equations for lines in space"

Transcription

1 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 written with an arrow over the top such as v or in angle brackets such as v 1, v 2, v 3 indicates a vector (we may also refer to this vector using the notation v 1 i + v 2 j + v 3 k) Any quantity written in plain notation like v or in parenthesis such as (v 1, v 2, v 3 ) indicates a scalar or a point (unless otherwise noted) A quantity with a bar over the top, like v, indicates a line segment Standard equations for lines in space We can completely describe a line by specifying (1) a point through which the line passes, and (2) the line s direction In two dimensions, the line s direction is given by its slope In three dimensions, we will specify the line s direction using a vector parallel to the line Suppose that the line L passes through the point r 0 = (x 0, y 0, z 0 ) parallel to vector v = a, b, c (with terminal point v = (a, b, c)) We can move v so that it lies along the line by adding r 0 to vs initial and terminal points; the vector with initial point r 0 and terminal point r 0 + v = (a + x 0, b + y 0, c + z 0 ) lies on L: By stretching or shrinking v and adding this vector to r 0, we can reach any point on L Since stretching and shrinking a vector corresponds to scaling the vector (multiplying the vector by a scalar), any point on the line L may be written as r 0 + tv for some scalar t These observations give us a way to write an equation for the line: 1

2 Vector equation for a line: If the line L passes through the point r 0 = (x 0, y 0, z 0 ) (with position vector r 0 = x 0, y 0, z 0 ) parallel to the vector v = a, b, c, then the line has vector equation r(t) = r 0 + t v, where < t < The numbers a, b, and c are called the direction numbers of the line L Notice that r(t) is a vector; so the vector equation for L actually gives us position vectors whose terminal points lie on L We may also describe the line using the components x, y, and z: Parametric equations for a line: If the line L passes through the point r 0 = (x 0, y 0, z 0 ) (with position vector r 0 = x 0, y 0, z 0 ) parallel to the vector v = a, b, c, then the line may be described by the equations where < t < x = x 0 + ta, y = y 0 + tb, z = z 0 + tc, Another method for specifying the line L is by using symmetric equations If L has parametric equations x = x 0 + ta, y = y 0 + tb, and z = z 0 + tc, then by solving for t in each equation, we have t = x x 0 a, t = y y 0 b, and t = z z 0, c assuming that each of a, b, or c is nonzero Since t is the same in each equation, we have the following description of L: Symmetric equations for a line: If the line L passes through the point r 0 = (x 0, y 0, z 0 ) (with position vector r 0 = x 0, y 0, z 0 ) parallel to the vector v = a, b, c, then the line may be described by the equations x x 0 a = y y 0 b = z z 0 c Example: Find the vector equation and parametric equations for the line passing through the points p 1 = (1, 3, 1) and p 2 = (7, 3, 6) Then paramaterize the line segment p 1 p 2 The two points define a vector p 2 p 1 = 7 1, 3 3, = 6, 6, 5 which is parallel to the line passing through p 1 and p 2 So a vector equation for the line is r(t) = 1, 3, 1 + t 6, 6, 5, < t < Note that we could have written the equation for the line in many different ways; 1, 3, 1 + s 6, 6, 5, < s < 7, 3, 6 + p 6, 6, 5, < p < 2

3 and 7, 3, 6 + q 6, 6, 5, < q < are all equivalent vector equations for this line Parametric equations for the line are given by x = 1 + 6t, y = 3 6t, z = 1 5t, < t < Again, there are many different ways to write equivalent conditions for the line To parameterize the segment from p 1 to p 2, we may use the same parametric equations for the line itself, but we must put bounds on t so that we only describe points on the segment from p 1 to p 2 We need to determine the smallest allowable value for t (which will give us the point p 1 ), and the largest value for t, which will give us p 2 Using the x coordinate of the point p 1, if 1 + 6t = 1, then t = 0 It is easy to see that plugging 0 into each coordinate returns the point p 1 So the smallest value for t that we can allow is t = 0 Using the x coordinate of p 2, suppose that 1+6t = 7 Then t = 1 (and again we see that plugging 1 into each equation will indeed return the point p 2 ) So the equations for the line segment p 1 p 2 are x = 1 + 6t, y = 3 6t, z = 1 5t, 0 < t < 1 Distance from a point to a line To measure the distance from a point p 0 to a line L, we look for the point l on L closest to p 0, then determine the distance from l to p 0 In particular, the segment joining l and p 0 should be perpendicular to L In practice, given any point p on L, the process outlined above can be streamlined using some geometry We want to find the length of the side opposite θ in the right triangle below: 3

4 The length of the hypotenuse is pp 0, so the length of the sides are pp 0 sin θ and pp 0 cos θ In particular, the opposite side has length pp 0 sin θ Recall that the length of the cross product of pp 0 with v is pp 0 v = pp 0 v sin θ Rewriting the above equation as pp 0 v = pp 0 sin θ v gives us a method for finding the length pp 0 sin θ: we can instead calculate pp 0 v v Theorem 001 If the point p lies on the line L, p 0 is any point not on L, and v is a vector parallel to L, then the distance from p 0 to L is given by pp 0 v v Example: Determine the distance from the point p 0 = ( 2, 4, 1) to the line L with vector equation r(t) = 3, 0, 0 + t 4, 1, 1 In order to use the formula pp 0 v, v 4

5 we must know a point p on the line L and a vector v parallel to L Fortunately, it is easy to find such a v Since L has vector equation the vector r(t) = 3, 0, 0 + t 4, 1, 1, v = 4, 1, 1 is parallel to L We can find a point on L by plugging a value for t into t = 0 Then r(0) = 3, 0, , 1, 1 = 3, 0, 0 is a vector whose terminal point lies on L So the point p = (3, 0, 0) is on L The vector pp 0 is given by pp 0 = 5, 4, 1 So to determine the distance from p 0 to L, we must calculate pp 0 v 5, 4, 1 4, 1, 1 = v 4, 1, 1 5, 4, 1 4, 1, 1 = = 5 i + j + 21 k 18 r(t); for ease of computation, let s let = = units Equation for a plane In order to completely describe a plane P, we must specify (1) a point p 0 that lies on the plane, as well as (2) the tilt of the plane We talk about the plane s tilt or direction using any vector n normal (ie orthogonal) to the plane Notice that, if p 0 is a point on the plane P with normal vector n, then 5

6 every vector that lies on the plane P is orthogonal to n, and the point p lies on the plane P if and only if the vector pp 0 is orthogonal to n Since vectors are orthogonal if and only if their dot product is 0, we may reformulate the latter idea: point p lies on the plane P if and only if pp 0 n = 0 Let us make the arguments more precise Suppose that a normal vector of the plane P is given by n = a i + b j + c k, and that the point r 0 = (x 0, y 0, z 0 ) lies on P With r = (x, y, z), the dot product rr 0 n is given by rr 0 n = x x 0, y y 0, z z 0 a, b, c = a(x x 0 ) + b(y y 0 ) + c(z z 0 ) Then the point r lies on P if and only if rr 0 n = a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0 So P is the set of all points (x, y, z) so that rr 0 n = n 1 (x x 0 ) + n 2 (y y 0 ) + n 3 (z z 0 ) = 0 Theorem 002 If P is a plane passing through the point r 0 = (x 0, y 0, z 0 ) with normal vector n = a i + b j + c k, then P is the set of all points r = (x, y, z) so that (vector equation) n rr 0 = 0 (scalar equation) a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0 (modified scalar equation) ax + by + cz = ax 0 + by 0 + cz 0 Examples Find the equation for the plane P passing through the point p = (2, 1, 0) and parallel to the plane P 0 defined by the equation 3x 7y + 4z = 12 Since the plane we want is parallel to the plane 3x 7y + 4z = 12, the two planes have the same normal vector Now P 0 has normal vector 3, 7, 4, so using the form a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0, P is defined by 3(x x 0 ) 7(y y 0 ) + 4(z z 0 ) = 0, 6

7 or 3(x 2) 7(y 1) + 4(z 0) = 0; we may rewrite this equation as 3x 7y + 4z = 1 Find the equation for the plane P passing through the points (4, 0, 3) and (2, 1, 1), and parallel to the line L with parametric equations x = 1 + t, y = 3 3t, z = 12t In order to use the equation a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0, we need to find a vector normal to P The plane must be parallel to the line L, which in turn is parallel to the vector v = 1, 3, 12 Since the cross product of a pair of (nonparallel) vectors is normal to both of the original vectors, the cross product of v with any other (nonparallel) vector on P will be a vector normal to P So we need to find another vector on P Since we know two points on P, this is easy to accomplish: u = 2, 1, 2 is parallel to P Then v and u have cross product v u = 1, 3, 12 2, 1, 2 = 6 i + 22 j + 5 k, and this vector is normal to P So we may write the equation for P as 6(x 4) + 22(y 0) + 5(z 3) = 0, or 6x + 22y + 5z = 39 Lines of Intersection A pair of lines is parallel if they have the same slope A pair of planes is parallel if they have the same tilt In 3 dimensions, we measure the tilt of the plane using vectors normal to the plane; so if planes N and M are parallel, there normal vectors must also be parallel If planes N and M are not parallel, then the planes intersect in a line L that lies on both planes; thus L must be perpendicular to the normal vectors of each plane Example Find an equation for the line of intersection of the planes 3x 4y+12z = 14 and x+2y z = 3 Recall that, if we are to write an equation for a line L in 3 dimensional space, we must find a point p on the line and a vector v parallel to the line The vector v is not difficult to find; L is perpendicular to the normal vectors of each plane, so the cross product of the normal vectors will be a vector parallel to L 7

8 We calculate v using the cross product formula: i j k v = = i(4 24) j( ) + k(6 4) = 20 i 9 j + 2 k Now that we have a vector parallel to the line of intersection of the two planes, we need to find a point on the line of intersection A point on the line of intersection is a point (x 0, y 0, z 0 ) that lies on both planes In other words, this point (x 0, y 0, z 0 ) should satisfy both equations 3x 4y + 12z = 14 and x + 2y z = 3, ie 3x 0 4y z 0 = 14 and x 0 + 2y 0 z 0 = 3 We currently have two equations with three unknowns, but we can reduce the problem to one we can solve (two equations with two unknowns) by making a choice for one variable Since the x value for the line of intersection is clearly not constant, there is some point on the line where x 0 = 0 Let s determine the values for y 0 and z 0 where this occurs If x 0 = 0, we have 4y z 0 = 14 and 2y 0 z 0 = 3 Let s solve this system of equations: Multiply the second line by 2: Now add the two equations to eliminate y 0 : 4y z 0 = 14 2y 0 z 0 = 3 4y z 0 = 14 4y 0 2z 0 = 6 4y z 0 = y 0 2z 0 = 6 10z 0 = 20 So z 0 = 2 Substituting 2 into the second equation and solving for y 0, we have 2y 0 2 = 3 2y 0 = 5 y 0 = 5 2 So the point ( 0, 5 2, 2) is on the line of intersection Thus the parametric equations for L are given by x = 20t, y = 5 9t, z = 2 + 2t 2 Distance from a point to a plane 8

9 To measure the distance from a point p to a plane P, we will compare the point to the plane s normal vector n: if the initial point of n is p 0, then we may measure the distance from the point p to the plane along the normal vector n by projecting the vector p 0 p onto n Recall from 123 that the length of vector projection of p 0 p onto n is given by proj n p 0 p = n p 0p n So the distance from p to the plane with point p 0 and normal vector n is n p 0 p n Example Find the distance from the point p = ( 2, 1, 5) to the plane P with equation 5x 12y + z = 7 P has normal vector n = 5, 12, 1 ; we will also need a point on the plane; p 0 = (0, 0, 7) will work nicely Finally we need to find the length of the projection of the vector p 0 p = 2, 1, 2 onto n So the distance is given by n p 0 p p 0 p = = 5, 12, 1 2, 1, = Angles between planes The easiest way to find the angle of intersection of two planes P 1 and P 2 is by noting that the angle of intersection of their normal vectors n 1 and n 2 is precisely the same as the angle of intersection of the planes 9

10 In particular, the angle θ between vectors n 1 and n 2 can be calculated using the dot product From section 123, we know that angle θ between vectors n 1 and n 2 is ( ) θ = cos 1 n1 n 2 n 1 n 2 Theorem 003 If planes P 1 and P 2 have normal vectors n 1 and n 2 respectively, then the angle θ between P 1 and P 2 is given by ( ) θ = cos 1 n1 n 2 n 1 n 2 10

Section 9.5: Equations of Lines and Planes

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 # 3-5 odd, 2-37 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that

More information

Math 241, Exam 1 Information.

Math 241, Exam 1 Information. Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

More information

LINES AND PLANES CHRIS JOHNSON

LINES AND PLANES CHRIS JOHNSON LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3-space, as well as define the angle between two non-parallel planes, and determine the distance

More information

= y y 0. = z z 0. (a) Find a parametric vector equation for L. (b) Find parametric (scalar) equations for L.

= 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 information

Section 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables

Section 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 information

Lecture 14: Section 3.3

Lecture 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 information

12.5 Equations of Lines and Planes

12.5 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

9 Multiplication of Vectors: The Scalar or Dot Product

9 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 information

1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero vectors u and v,

1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero 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 information

A vector is a directed line segment used to represent a vector quantity.

A 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 information

Section 13.5 Equations of Lines and Planes

Section 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 information

MAT 1341: REVIEW II SANGHOON BAEK

MAT 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 information

Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors

Adding 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 information

1.5 Equations of Lines and Planes in 3-D

1.5 Equations of Lines and Planes in 3-D 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 3-D Recall that given a point P = (a, b, c), one can draw a vector from

More information

Section 2.4: Equations of Lines and Planes

Section 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 information

L 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

L 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 information

Section 1.1. Introduction to R n

Section 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

THREE DIMENSIONAL GEOMETRY

THREE 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 information

10.5. 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.

10.5. 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 information

FURTHER VECTORS (MEI)

FURTHER 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: -9-7 Mathematics

More information

Section 11.4: Equations of Lines and Planes

Section 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 information

5.3 The Cross Product in R 3

5.3 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 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,

(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 information

JUST 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) 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 information

Review Sheet for Test 1

Review Sheet for Test 1 Review Sheet for Test 1 Math 261-00 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 information

11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space

11.1. 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 information

Section 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.

Section 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 information

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

Linear 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 information

13.4 THE CROSS PRODUCT

13.4 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

Solutions to old Exam 1 problems

Solutions 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 information

a.) Write the line 2x - 4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a

a.) 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 information

Mathematics 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. 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 information

Lines and Planes in R 3

Lines 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 information

Determine 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

Determine 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

28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z

28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z 28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition

More information

2.1 Three Dimensional Curves and Surfaces

2.1 Three Dimensional Curves and Surfaces . Three Dimensional Curves and Surfaces.. Parametric Equation of a Line An line in two- or three-dimensional space can be uniquel specified b a point on the line and a vector parallel to the line. The

More information

CHAPTER FIVE. 5. Equations of Lines in R 3

CHAPTER 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 information

Orthogonal Projections and Orthonormal Bases

Orthogonal Projections and Orthonormal Bases CS 3, HANDOUT -A, 3 November 04 (adjusted on 7 November 04) Orthogonal Projections and Orthonormal Bases (continuation of Handout 07 of 6 September 04) Definition (Orthogonality, length, unit vectors).

More information

Vector Algebra CHAPTER 13. Ü13.1. Basic Concepts

Vector 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 information

Math 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t

Math 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 information

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

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 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 information

Two vectors are equal if they have the same length and direction. They do not

Two 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 information

The Dot and Cross Products

The 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 information

LINES AND PLANES IN R 3

LINES 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 information

discuss how to describe points, lines and planes in 3 space.

discuss 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 information

Figure 1.1 Vector A and Vector F

Figure 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 information

α = u v. In other words, Orthogonal Projection

α = 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 information

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

1.(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

Definition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.

Definition: 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 information

Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155

Name 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 information

Math 215 HW #6 Solutions

Math 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 information

Solutions to Exercises, Section 5.1

Solutions 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 information

A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS

A 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 information

Equations of Lines and Planes

Equations 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 information

Some Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)

Some Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000) Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving

More information

C relative to O being abc,, respectively, then b a c.

C relative to O being abc,, respectively, then b a c. 2 EP-Program - Strisuksa School - Roi-et 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 information

Exam 1 Sample Question SOLUTIONS. y = 2x

Exam 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 information

MATH 275: Calculus III. Lecture Notes by Angel V. Kumchev

MATH 275: Calculus III. Lecture Notes by Angel V. Kumchev MATH 275: Calculus III Lecture Notes by Angel V. Kumchev Contents Preface.............................................. iii Lecture 1. Three-Dimensional Coordinate Systems..................... 1 Lecture

More information

6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu)

6. Vectors. 1 2009-2016 Scott Surgent (surgent@asu.edu) 6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,

More information

Geometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi

Geometry 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 three-dimensional space, we also examine the

More information

Dot 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) 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 information

Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 25 Notes

Jim Lambers MAT 169 Fall Semester 2009-10 Lecture 25 Notes Jim Lambers MAT 169 Fall Semester 009-10 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 information

i=(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.

i=(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 information

Lines and Planes 1. x(t) = at + b y(t) = ct + d

Lines and Planes 1. x(t) = at + b y(t) = ct + d 1 Lines in the Plane Lines and Planes 1 Ever line of points L in R 2 can be epressed as the solution set for an equation of the form A + B = C. The equation is not unique for if we multipl both sides b

More information

Solutions to Math 51 First Exam January 29, 2015

Solutions 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 information

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

More information

Geometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures.

Geometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures. Geometry: Unit 1 Vocabulary 1.1 Undefined terms Cannot be defined by using other figures. Point A specific location. It has no dimension and is represented by a dot. Line Plane A connected straight path.

More information

Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

More information

9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes

9.4. 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 so-called because when the scalar product of

More information

v 1 v 3 u v = (( 1)4 (3)2, [1(4) ( 2)2], 1(3) ( 2)( 1)) = ( 10, 8, 1) (d) u (v w) = (u w)v (u v)w (Relationship between dot and cross product)

v 1 v 3 u v = (( 1)4 (3)2, [1(4) ( 2)2], 1(3) ( 2)( 1)) = ( 10, 8, 1) (d) u (v w) = (u w)v (u v)w (Relationship between dot and cross product) 0.1 Cross Product The dot product of two vectors is a scalar, a number in R. Next we will define the cross product of two vectors in 3-space. This time the outcome will be a vector in 3-space. Definition

More information

... ... . (2,4,5).. ...

... ... . (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 information

LINEAR ALGEBRA W W L CHEN

LINEAR 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 information

Section 12.6: Directional Derivatives and the Gradient Vector

Section 12.6: Directional Derivatives and the Gradient Vector Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate

More information

Vector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.

Vector 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 information

Vector Math Computer Graphics Scott D. Anderson

Vector 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 information

3. INNER PRODUCT SPACES

3. 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 information

PYTHAGOREAN TRIPLES KEITH CONRAD

PYTHAGOREAN TRIPLES KEITH CONRAD PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

More information

VECTOR ALGEBRA. 10.1.1 A quantity that has magnitude as well as direction is called a vector. is given by a and is represented by a.

VECTOR ALGEBRA. 10.1.1 A quantity that has magnitude as well as direction is called a vector. is given by a and is represented by a. VECTOR ALGEBRA Chapter 10 101 Overview 1011 A quantity that has magnitude as well as direction is called a vector 101 The unit vector in the direction of a a is given y a and is represented y a 101 Position

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

Lesson 19: Equations for Tangent Lines to Circles

Lesson 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

One advantage of this algebraic approach is that we can write down

One advantage of this algebraic approach is that we can write down . Vectors and the dot product A vector v in R 3 is an arrow. It has a direction and a length (aka the magnitude), but the position is not important. Given a coordinate axis, where the x-axis points out

More information

MA261-A Calculus III 2006 Fall Homework 3 Solutions Due 9/22/2006 8:00AM

MA261-A Calculus III 2006 Fall Homework 3 Solutions Due 9/22/2006 8:00AM MA6-A 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)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

Solutions to Homework 5

Solutions to Homework 5 Solutions to Homework 5 1. Let z = f(x, y) be a twice continously differentiable function of x and y. Let x = r cos θ and y = r sin θ be the equations which transform polar coordinates into rectangular

More information

Recall that two vectors in are perpendicular or orthogonal provided that their dot

Recall 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 information

Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.

Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular. CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes

More information

Problem set on Cross Product

Problem 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 information

Geometry and Measurement

Geometry and Measurement The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

More information

ISOMETRIES OF R n KEITH CONRAD

ISOMETRIES OF R n KEITH CONRAD ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x

More information

Unified Lecture # 4 Vectors

Unified Lecture # 4 Vectors Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,

More information

Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.

Problem 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 information

[1] Diagonal factorization

[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 information

Section 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 25-37, 40, 42, 44, 45, 47, 50

Section 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 25-37, 40, 42, 44, 45, 47, 50 Section 11.1: Vectors in the Plane Page 779 Suggested Problems: 1, 5, 9, 17, 3, 5-37, 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 information

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

More information

Cross 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 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 information

Solutions for Review Problems

Solutions 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 information

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions. Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

More information

Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

More information

Vector 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 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 information