Determinants, Areas and Volumes

Size: px
Start display at page:

Download "Determinants, Areas and Volumes"

Transcription

1 Determinants, Areas and Volumes Theodore Voronov Part 2 Areas and Volumes The area of a two-dimensional object such as a region of the plane and the volume of a three-dimensional object such as a solid body in space, as well the length of an interval of the real line, are all particular cases of a very general notion of measure. General measure theory is a part of analysis. Here we shall focus on the geometrical idea of measure and its relation with the algebraic notion of determinant. Why the area of a parallelogram is represented by a determinant Although, from practice we know very well what is the area of simple geometrical figures such as, for example, a rectangle or a disk, it is not easy to give a rigorous general definition of area. However, some properties of area should be clear; consider subsets of the plane R 2 : (1) Area is always non-negative: area(s) 0 for all subsets S R 2 such that it makes sense to speak about their area; (2) Area is additive: area(s 1 S 2 ) = areas 1 + areas 2, if the intersection S 1 S 2 is empty. A translation of the space R n is the map T a : R n R n that takes every point x R n to x + a, where a R n is a fixed vector. For a subset S R n, a translation T a shifts all points of S along a, i.e., each point x S is mapped to x + a, and S moves rigidly to its new location in R n. The following properties hold for areas in R 2 : (3) Area is invariant under translations: areat a (S) = areas, for all vectors a R 2. (4) For a one-dimensional object, such as a line segment, the area should vanish. The plan now is as follows: using conditions (1) (4) we shall establish a deep link between the notion of area and the theory of determinants. To this end, we consider the area of a simple polygon, a parallelogram. Later our considerations will be generalized to R n. 1

2 Let a = (a 1, a 2 ), b = (b 1, b 2 ) be vectors in R 2. The parallelogram on a, b with basepoint O R 2 is the set of points of the form x = O + ta + sb where 0 t, s 1. One can easily see that it is the plane region bounded by the two pairs of parallel straight line segments: OA, BC and OB, AC where C = O + a + b. b B C O a A What is the area of it? From property (3) it follows that the area does not depend on the location of our parallelogram in R 2 : by a translation the basepoint O can be made an arbitrary point of the plane without changing the area. Let us assume that O = 0 is the point (0, 0). Denote the parallelogram by Π(a, b). Then areaπ(a, b) is a function of vectors a, b. Proposition 1 The function areaπ(a, b) has the following properties: (1) areaπ(na, b)= areaπ(a, nb)= n areaπ(a, b) for any n Z; (2) areaπ(a, b + ka)= areaπ(a + kb, b)= areaπ(a, b) for any k R. Proof Suppose we replace a by na for a positive integer n. Then Π(na, b) is the union of n copies of the parallelogram Π(a, b): b b a b a a From the additivity of area it follows that areaπ(na, b)= n Π(a, b). The same is true if we replace b by nb, for positive n. For n = 0, Π(0, b) or Π(a, 0) are just segments, therefore have zero area, by (4). Notice also that Π( a, b) and Π(a, b) differ by a shift, so have the same area. Hence, areaπ(na, b)= areaπ(a, nb)= n areaπ(a, b) holds in general. To prove the second relation, we again use the additivity of area: it is clear that to obtain Π(a, b + ka), one has to cut from Π(a, b) the triangle OBB, shift it by the vector a and attach it back as the triangle ACC : B B C C b O a A 2

3 Clearly, due to additivity and invariance under translations, the area will not change, and we arrive at areaπ(a, b + ka)= areaπ(a, b) as claimed. In fact, the first assertion in Proposition 1 is valid in a stronger form. Proposition 2 The area of a parallelogram satisfies for any real number k R. Proof Omitted. areaπ(ka, b)= areaπ(a, kb)= k areaπ(a, b) Theorem 1 Suppose a= (a 1, a 2 ), b= (b 1, b 2 ). Then ( ) areaπ(a, b) = det(a, b) = det a1 a 2. (1) b 1 b 2 Proof Indeed, by Propositions 1 and 2, areaπ(a, b) satisfies almost the same properties as the determinant det(a, b). They can be used to calculate areaπ(a, b); it is similar to using row operations for calculating determinants. We have ( ) a1 a 2 ( a1 a = areaπ 2 = a 1 b 2 b 1 a 1 a 2 areaπ where e 1 = (1, 0) and e 2 = (0, 1), as desired. areaπ(a, b)= areaπ ) b 1 ( b 2 ) a1 0 0 b 2 b = areaπ 1 a 1 a 2 0 b 2 b 1 ( ) a 1 a = a b 2 b 1 a 2 areaπ(e 1, e 2 ) = a 1 b 2 b 1 a 2, N.B. In the theorem we chose the unit of area so that areaπ(e 1, e 2 ) = 1. Example Find the area of the parallelogram built on vectors a = ( 2, 5) and b = (1, 1). Solution: we have = = 7. Hence areaπ(a, b) = 7 = 7. Example Find the area of the triangle ABC if A = (3, 2), B = (4, 2), C = (1, 0). Solution: it is half of the area of the parallelogram built on CA = A C = (2, 2), CB = B C = (3, 2). Hence area(abc) = = 1. 3

4 Volumes and determinants All the above results can be generalized to higher dimensions. Consider vectors a 1,..., a n in R n. The parallelepiped on a 1,..., a n with basepoint O R n is the set of points x = O+t 1 a t n a n where 0 t 1 1 for all i = 1,..., n. Denote it Π(a 1,..., a n ). In the sequel nothing depends on the basepoint, so we shall suppress any mentioning of it. Instead of deducing a formula for the volume of a parallelepiped similar to (1) from general properties of volumes such as additivity, as we did above for area, it is convenient to set by definition a a 1n volπ(a 1,..., a n ) = (2) a n1... a nn where a 1 = (a 11,..., a 1n ),..., a n = (a n1,..., a nn ). Note that if this is negative we take the absolute value. This definition implies that the unit of volume is such that the volume of Π(e 1,..., e n ) is set to 1. Example Find the oriented volume of the parallelepiped built on a 1 = (2, 1, 0), a 2 = (0, 3, 11) and a 3 = (1, 2, 7) in R 3. Solution: volπ(a 1, a 2, a 3 ) = = 2 ( 1) 1 ( 11) = So the volume equals 9. Areas and volumes in Euclidean space Recall that on R n one can define the scalar product of vectors by the formula (a, b)= a 1 b a n b n (see Problem 5 in week 1. We can also write a b). It follows that the standard basis vectors e 1,..., e n satisfy (e i, e j ) = 0 if i j and (e i, e i ) = 1. The length of a vector is defined as a = (a, a) and the angle between two vectors is defined by the equality (a, b) = a b cos α 4

5 from where we can find cos α if (a, b), (a, a), (b, b) are known. The unit cube Π(e 1,..., e n ) plays above a distinguished role. We shall see that any unit cube in R n will have unit volume. Consider an example (for n = 2 we continue to use area instead of vol). Example Let g 1 = (cos α, sin α), g 2 = ( sin α, cos α) in R 2. We can immediately see that g 1 = g 2 = 1 and g 1 g 2 = 0, so Π(g 1, g 2 ) is a unit square. We have areaπ(g 1, g 2 ) = cos α sin α sin α cos α = cos2 α + sin 2 α = 1. There is a way of expressing the volume of a parallelepiped entirely in terms of intrinsic geometric information: lengths of vectors and angles between them, rather than their coordinates as in the previous formulas. Definition G(a 1,..., a k ) = (a 1, a 1 )... (a 1, a k ) (3) (a k, a 1 )... (a k, a k ) where k n, is called the Gram matrix of the system of vectors a 1,..., a n and its determinant, the Gram determinant. Theorem 2 The Gram determinant of a 1,..., a n is the square of the volume of the parallelepiped Π(a 1,..., a n ). Proof Indeed, consider the n n matrix A with rows a i. Consider AA T = a a 1n a a n = a 1... ( a T 1... a T n ) a n1... a nn a 1n... a nn a n Hence = a 1a T 1... a 1 a T n a n a T 1... a n a T n = G(a 1,..., a n ). det G(a 1,..., a n )= det(aa T ) = det A det A ( T 2. = (det A) 2 = volπ(a 1,..., a n )) Corollary If g i are arbitrary mutually orthogonal unit vectors (i.e., Π(g 1,..., g n ) is a unit cube), then volπ(g 1,..., g n )= 1. 5

6 One of the advantages of expressing volumes via the Gram determinants is that it allows us to consider easily parallelepipeds in R n of dimensions less than n. More precisely, if we are given k vectors a 1,..., a k, then we can consider a k-dimensional parallelepiped Π(a 1,..., a k ) contained in the k-dimensional space spanned by a 1,..., a k. The formula ( volπ(a 1,..., a k )) 2 = det G(a1,..., a k ) is applicable, with the scalar products calculated in the ambient space R n. Example Find the area of the parallelogram built on a= (1, 1, 2) and b= (2, 0, 3). Solution: the Gram determinant is = 14. Hence the area is 14. 6

7 MT1000 Project 4 Groupwork Week 2 Problem 1 Find the areas and volumes: (a) areaπ(a, b) if a= ( cos α, sin α), b= ( sin α, cos α) in R 2 ; make a sketch; (b) volπ(a, b, c) if a= (3, 2, 1), b= (2, 2, 5), c= (0, 0, 1) in R 3 ; (c) areaπ(a, b) if a= (1, 1, 2, 3), b= (0, 3, 1, 2) in R 4. (Use the Gram determinant.) Problem 2 Verify by a direct calculation that the Gram determinant det G(a, b) vanishes if one of the vectors a, b is a scalar multiple of the other. (Geometrically that means that they are in the same line.) What can be said about the area of the parallelogram Π(a, b)? Problem 3 Show that the oriented area of a triangle ABC in R 2 is given by the formula a 1 a 2 1 area(abc) = 1 2 b 1 b 2 1 c 1 c 2 1 where A = (a 1, a 2 ), B = (b 1, b 2 ), C = (c 1, c 2 ). Problem 4 (See Problem 5 from week 1) (a) Show that the so-called triple or mixed product (a, b, c) in R 3 defined as (a, b, c) = a (b c) = (a b) c is the volume of Π(a, b, c). (b) Let n be a unit vector perpendicular to a and b. Show that det G(n, a, b)= det G(a, b) and deduce that volπ(n, a, b)= areaπ(a, b). (c) Use parts (a) and (b) to prove that the length of the vector product a b in R 3 is the area of the parallelogram Π(a, b). Problem 5 Show that a b = a b sin α where α is the angle between a and b. Hint: use the result of the previous problem and express the area by the Gram determinant. You may assume that a b = a b cos α. Project Report You should write a report on the topics covered in this project. The report should include a description (in your own words) of the defining properties of a determinant and the key properties that follow from this (you do not need to include properties of matrices and vectors). The link between determinants and areas and volumes should be explained and also the role of the Gram determinant in calculating volumes (do not include the proofs of Proposition 1 and Theorems 1 and 2). The solutions to all groupwork problems should be included in an 7

8 appropriate place. Even though you have worked as a group on this project, the report should be all your own work. Please hand in your report (with your name and group number on the front) to the Student Support Office in Lamb by 1pm on Friday 16th December. There are 25 marks for this project. (a) 10 marks for the solutions to the mathematical problems. (b) 10 marks for clearly and correctly explaining the key ideas in the lecture notes in your own words. (c) half the average mark for your group for (a) out of 5. Any student who does not attend a group session, without good reason, will get 50% of the group mark (c). Any student who does not attend both group sessions, without good reason, will get 0 for the group mark. Please notify us as soon as possible if you miss a session and fill in a Self Certification Form available from the Student Support Office. 8

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

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

Notice that v v w (4)( 15) ( 3)( 20) (0)(2) ( 2)( 15) (2)( 20) (5)(2)

Notice 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 information

MATHEMATICAL THEORY FOR SOCIAL SCIENTISTS MATRIX ALGEBRA

MATHEMATICAL THEORY FOR SOCIAL SCIENTISTS MATRIX ALGEBRA MATRIX ALGEBRA By gathering the elements of the equation B A 1 under (16) and (17), we derive the following expression for the inverse of a 2 2 matrix: (18) 1 a11 a 12 1 a22 a 12 a 21 a 22 a 11 a 22 a

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

Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,

Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n, LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x

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

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

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

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

Matrices Gaussian elimination Determinants. Graphics 2011/2012, 4th quarter. Lecture 4: matrices, determinants

Matrices Gaussian elimination Determinants. Graphics 2011/2012, 4th quarter. Lecture 4: matrices, determinants Lecture 4 Matrices, determinants m n matrices Matrices Definitions Addition and subtraction Multiplication Transpose and inverse a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn is called an m n

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

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

v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.

v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed 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 information

Inner products and orthogonality

Inner products and orthogonality Chapter 5 Inner products and orthogonality Inner product spaces, norms, orthogonality, Gram-Schmidt process Reading The list below gives examples of relevant reading. (For full publication details, see

More information

VECTORS IN THREE DIMENSIONS

VECTORS IN THREE DIMENSIONS 1 CHAPTER 2. VECTORS IN THREE DIMENSIONS 1 INSTITIÚID TEICNEOLAÍOCHTA CHEATHARLACH INSTITUTE OF TECHNOLOGY CARLOW VECTORS IN THREE DIMENSIONS 1 Vectors in Three Dimensions y (u 1, u 2, u 3 ) u O z x 1.1

More information

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:

Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible

More information

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

1.3. DOT PRODUCT If θ is the angle (between 0 and π) between two non-zero vectors u and v, 1.3. DOT PRODUCT 29 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

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

After studying this unit, you should be able to: define scalar product or dot product of vectors; find angle between two vectors;

After studying this unit, you should be able to: define scalar product or dot product of vectors; find angle between two vectors; Vectors and Three Dimensional Geometry UNIT 2 VECTORS 2 Structure 2.0 Introduction 2.1 Objectives 2.2 Scalar Product of Vectors 2.3 Vector Product (or Cross Product) of two Vectors 2.4 Triple Product of

More information

MAC 2313 VECTORS, CONIC SECTIONS, DOT PRODUCT

MAC 2313 VECTORS, CONIC SECTIONS, DOT PRODUCT MAC 2313 VECTORS, CONIC SECTIONS, DOT PRODUCT I. Vectors. In this section, you will use triangle and/or parallelogram law for adding vectors together with basic properties of addition and scalar multiplication.

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

We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P. Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to

More information

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

More information

MATH 304 Linear Algebra Lecture 11: Basis and dimension.

MATH 304 Linear Algebra Lecture 11: Basis and dimension. MATH 304 Linear Algebra Lecture 11: Basis and dimension. Linear independence Definition. Let V be a vector space. Vectors v 1,v 2,...,v k V are called linearly dependent if they satisfy a relation r 1

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

13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.

13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions. 3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms

More information

Course 2BA1: Michaelmas Term 2006 Section 6: Vectors

Course 2BA1: Michaelmas Term 2006 Section 6: Vectors Course 2BA1: Michaelmas Term 2006 Section 6: Vectors David R. Wilkins Copyright c David R. Wilkins 2005 06 Contents 6 Vectors 1 6.1 Displacement Vectors....................... 2 6.2 The Parallelogram Law

More information

Vector Spaces Math 1553 Fall Ambar Sengupta

Vector Spaces Math 1553 Fall Ambar Sengupta Vector Spaces Math 1553 Fall 2009 Ambar Sengupta A vector space V is a set of objects, called vectors on which there are two operations defined: addition multiplication by scalar (v, w) v + w (k, v) kv

More information

Similarity and Diagonalization. Similar Matrices

Similarity and Diagonalization. Similar Matrices MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

More information

We know a formula for and some properties of the determinant. Now we see how the determinant can be used.

We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we

More information

Course MA2C02, Hilary Term 2012 Section 9: Vectors and Quaternions

Course MA2C02, Hilary Term 2012 Section 9: Vectors and Quaternions Course MAC0, Hilary Term 01 Section 9: Vectors and Quaternions David R. Wilkins Copyright c David R. Wilkins 000 01 Contents 9 Vectors and Quaternions 51 9.1 Vectors............................... 51 9.

More information

Linear Algebra II. Notes 1 September a(b + c) = ab + ac

Linear Algebra II. Notes 1 September a(b + c) = ab + ac MTH6140 Linear Algebra II Notes 1 September 2010 1 Vector spaces This course is about linear maps and bilinear forms on vector spaces, how we represent them by matrices, how we manipulate them, and what

More information

SUBSPACES. Chapter Introduction. 3.2 Subspaces of F n

SUBSPACES. Chapter Introduction. 3.2 Subspaces of F n Chapter 3 SUBSPACES 3. Introduction Throughout this chapter, we will be studying F n, the set of all n dimensional column vectors with components from a field F. We continue our study of matrices by considering

More information

Lecture L3 - Vectors, Matrices and Coordinate Transformations

Lecture L3 - Vectors, Matrices and Coordinate Transformations S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

More information

We thus get the natural generalization of the distance formula to three dimensions: 1

We thus get the natural generalization of the distance formula to three dimensions: 1 Three Dimensional Euclidean Space We set up a coordinate system in space (three dimensional Euclidean space) by adding third axis perpendicular to the two axes in the plane (two dimensional Euclidean space).

More information

MATH 304 Linear Algebra Lecture 16: Basis and dimension.

MATH 304 Linear Algebra Lecture 16: Basis and dimension. MATH 304 Linear Algebra Lecture 16: Basis and dimension. Basis Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis. Equivalently, a subset S V is a basis for

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

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

The Determinant: a Means to Calculate Volume

The Determinant: a Means to Calculate Volume The Determinant: a Means to Calculate Volume Bo Peng August 20, 2007 Abstract This paper gives a definition of the determinant and lists many of its well-known properties Volumes of parallelepipeds are

More information

Basic Linear Algebra. 2.1 Matrices and Vectors. Matrices. For example,, 1 2 3

Basic Linear Algebra. 2.1 Matrices and Vectors. Matrices. For example,, 1 2 3 Basic Linear Algebra In this chapter, we study the topics in linear algebra that will be needed in the rest of the book. We begin by discussing the building blocks of linear algebra: matrices and vectors.

More information

DOT and CROSS PRODUCTS

DOT and CROSS PRODUCTS DOT and CROSS PRODUCTS Vectors, whether in space or space, can be added, subtracted, scaled, and multiplied. There are two different products, one producing a scalar, the other a vector. Both, however,

More information

Vector Basics , 7, 1 2, 1 2

Vector Basics , 7, 1 2, 1 2 Vector Basics 1. Sketch each vector, assuming the initial point is at the origin. (a) 3, 4 (b) 2, 8 (c) 5, 0, 4 2. Calculate the magnitude of each of the vectors in Question 1. 3. The initial point P and

More information

1111: Linear Algebra I

1111: Linear Algebra I 1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Lecture 3 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 3 1 / 12 Vector product and volumes Theorem. For three 3D vectors u, v, and w,

More information

Review of vectors. The dot and cross products

Review of vectors. The dot and cross products Calculus 3 Lia Vas Review of vectors. The dot and cross products Review of vectors in two and three dimensions. A two-dimensional vector is an ordered pair a = a, a 2 of real numbers. The coordinate representation

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

Homework 1 Model Solution Section

Homework 1 Model Solution Section Homework Model Solution Section..4.... Find an equation of the sphere with center (,, 5) and radius 4. What is the intersection of this sphere with the yz-plane? Equation of the sphere: Intersection with

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

, we define the determinant of a 21 a 22 A, (also denoted by deta,) to be the scalar. det A = a 11 a 22 a 12 a 21.

, we define the determinant of a 21 a 22 A, (also denoted by deta,) to be the scalar. det A = a 11 a 22 a 12 a 21. 70 Chapter 4 DETERMINANTS [ ] a11 a DEFINITION 401 If A 12, we define the determinant of a 21 a 22 A, (also denoted by deta,) to be the scalar The notation a 11 a 12 a 21 a 22 det A a 11 a 22 a 12 a 21

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

(u, Av) = (A T u,v), (6.4)

(u, Av) = (A T u,v), (6.4) 216 SECTION 6.1 CHAPTER 6 6.1 Hermitian Operators HERMITIAN, ORTHOGONAL, AND UNITARY OPERATORS In Chapter 4, we saw advantages in using bases consisting of eigenvectors of linear operators in a number

More information

CHAPTER 5. VECTORS IN DIMENSION THREE OR HIGHER. Part 1. Vectors in 3D and the Cross Product

CHAPTER 5. VECTORS IN DIMENSION THREE OR HIGHER. Part 1. Vectors in 3D and the Cross Product CHAPTER 5. VECTORS IN DIMENSION THREE OR HIGHER Part 1. Vectors in 3D and the Cross Product Recall that the dot product (or inner product) u v of two planar vectors u = (u 1, u ) and v = (v 1, v ) is given

More information

HILBERT S AXIOM SYSTEM FOR PLANE GEOMETRY A SHORT INTRODUCTION

HILBERT S AXIOM SYSTEM FOR PLANE GEOMETRY A SHORT INTRODUCTION HILBERT S AXIOM SYSTEM FOR PLANE GEOMETRY A SHORT INTRODUCTION BJØRN JAHREN Euclid s Elements introduced the axiomatic method in geometry, and for more than 2000 years this was the main textbook for students

More information

MATH 304 Linear Algebra Lecture 22: Eigenvalues and eigenvectors (continued). Characteristic polynomial.

MATH 304 Linear Algebra Lecture 22: Eigenvalues and eigenvectors (continued). Characteristic polynomial. MATH 304 Linear Algebra Lecture 22: Eigenvalues and eigenvectors (continued). Characteristic polynomial. Eigenvalues and eigenvectors of a matrix Definition. Let A be an n n matrix. A number λ R is called

More information

DOT PRODUCT AND CROSS PRODUCT John P. D Angelo Dept. of Mathematics, Univ. of Illinois, 1409 W. Green St., Urbana IL

DOT PRODUCT AND CROSS PRODUCT John P. D Angelo Dept. of Mathematics, Univ. of Illinois, 1409 W. Green St., Urbana IL DOT PRODUCT AND CROSS PRODUCT John P. D Angelo Dept. of Mathematics, Univ. of Illinois, 1409 W. Green St., Urbana IL 61801 jpda@math.uiuc.edu The algebra of vectors What is a vector? In physics a vector

More information

Matrix Groups. Matrix Groups over Fields

Matrix Groups. Matrix Groups over Fields Matrix Groups Among the most important examples of groups are groups of matrices The textbook briefly discusses groups of matrices in Chapter 2 and then largely forgets about them These notes remedy this

More information

Unit 20 Linear Dependence and Independence

Unit 20 Linear Dependence and Independence Unit 20 Linear Dependence and Independence The idea of dimension is fairly intuitive. Consider any vector in R m, (a 1, a 2, a 3,..., a m ). Each of the m components is independent of the others. That

More information

Lecture 1 Introduction

Lecture 1 Introduction Tel Aviv University, Fall 2004 Lattices in Computer Science Lecture 1 Introduction Lecturer: Oded Regev Scribe: D. Sieradzki, V. Bronstein In this course we will consider mathematical objects known as

More information

Notes on Determinant

Notes on Determinant ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

More information

M427L Handout: Lines, Planes, Cross Products, & Coordinate Systems

M427L Handout: Lines, Planes, Cross Products, & Coordinate Systems M427L Handout: Lines, Planes, Cross Products, & Coordinate Systems Salman Butt June 14, 2007 Exercises (1) Compute (2, 9, 3) ( 1, 2, 6). (2) Compute v 1, v 2 and v 1 v 2 for v 1 = ( 1, 3, 1) and v 2 =

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

Mathematics Notes for Class 12 chapter 10. Vector Algebra

Mathematics 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 non-negative

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

1. Mathematical introduction Vector algebra. magnitude sense. direction. 1.1 Concept of vector. 1.2 Description of a vector with coordinates

1. Mathematical introduction Vector algebra. magnitude sense. direction. 1.1 Concept of vector. 1.2 Description of a vector with coordinates . Mathematical introduction Vector algebra. Concept of ector A ector is a mathematical object that has magnitude, direction and sense. Geometrically, e can picture a ector as a directed line segment, hose

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

12.3 The Dot Product. The Dot Product. The Dot Product. The Dot Product. Example 1. Vectors and the Geometry of Space

12.3 The Dot Product. The Dot Product. The Dot Product. The Dot Product. Example 1. Vectors and the Geometry of Space 12 Vectors and the Geometry of Space 12.3 Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. So far we have added two vectors and multiplied a vector by a

More information

Vector and Matrix Introduction

Vector and Matrix Introduction Vector and Matrix Introduction June 13, 2011 1 Vectors A scalar is a magnitude, or just a number An element of R is a scalar Until this point, most variables you have used denoted scalars Quantities measured

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

Determinants LECTURE Calculating the Area of a Parallelogram. Definition Let A be a 2 2 matrix. A = The determinant of A is the number

Determinants LECTURE Calculating the Area of a Parallelogram. Definition Let A be a 2 2 matrix. A = The determinant of A is the number LECTURE 13 Determinants 1. Calculating the Area of a Parallelogram Definition 13.1. Let A be a matrix. [ a c b d ] The determinant of A is the number det A) = ad bc Now consider the parallelogram formed

More information

1 Sets and Set Notation.

1 Sets and Set Notation. LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most

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

Basics from linear algebra

Basics from linear algebra Basics from linear algebra Definition. A vector space is a set V with the operations of addition + : V V V, denoted w + v = +( v, w), where v, w V and multiplication by a scalar : R V V denoted r v = (r,

More information

Numerical Analysis Lecture Notes

Numerical 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

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors Math 20F Linear Algebra Lecture 2 Eigenvalues and Eigenvectors Slide Review: Formula for the inverse matrix. Cramer s rule. Determinants, areas and volumes. Definition of eigenvalues and eigenvectors.

More information

7. The Gauss-Bonnet theorem

7. The Gauss-Bonnet theorem 7. The Gauss-Bonnet theorem 7. Hyperbolic polygons In Euclidean geometry, an n-sided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed

More information

CHAPTER 4. APPLICATIONS AND REVIEW IN TRIGONOMETRY

CHAPTER 4. APPLICATIONS AND REVIEW IN TRIGONOMETRY CHAPTER 4. APPLICATIONS AND REVIEW IN TRIGONOMETRY In the present chapter we apply the vector algebra and the basic properties of the dot product described in the last chapter to planar geometry and trigonometry.

More information

Geometry Performance Level Descriptors

Geometry Performance Level Descriptors Geometry Performance Level Descriptors Limited A student performing at the Limited Level demonstrates a minimal command of Ohio s Learning Standards for Geometry. A student at this level has an emerging

More information

MATH 12 LINEAR FUNCTIONS

MATH 12 LINEAR FUNCTIONS MATH 12 LINEAR FUNCTIONS ROSA ORELLANA 1. MULTIVARIABLE FUNCTIONS In this class we are interested in studying functions f : R n R m. This means that f assigns to every vector (or point) x = (x 1, x 2,...,

More information

Vectors 20/01/13. (above) Cartesian Coordinates (x,y,z) (above) Cartesian coordinates (x,y,z) and spherical coordinates (ρ,θ,φ).

Vectors 20/01/13. (above) Cartesian Coordinates (x,y,z) (above) Cartesian coordinates (x,y,z) and spherical coordinates (ρ,θ,φ). Vectors 0/01/13 Coordinate Systems We will work with a right handed Cartesian coordinate system labelled (x,y,z) for 3 dimensional Euclidean space R 3. Right handed means the axes are labeled according

More information

4 Determinant. Properties

4 Determinant. Properties 4 Determinant. Properties Let me start with a system of two linear equation: a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2. I multiply the first equation by a 22 second by a 12 and subtract the second

More information

Linear algebra vectors, matrices, determinants

Linear algebra vectors, matrices, determinants Linear algebra vectors, matrices, determinants Mathematics FRDIS MENDELU Simona Fišnarová Brno 2012 Vectors in R n Definition (Vectors in R n ) By R n we denote the set of all ordered n-tuples of real

More information

2.5 Spaces of a Matrix and Dimension

2.5 Spaces of a Matrix and Dimension 38 CHAPTER. MORE LINEAR ALGEBRA.5 Spaces of a Matrix and Dimension MATH 94 SPRING 98 PRELIM # 3.5. a) Let C[, ] denote the space of continuous function defined on the interval [, ] (i.e. f(x) is a member

More information

Linear Algebra for the Sciences. Thomas Kappeler, Riccardo Montalto

Linear Algebra for the Sciences. Thomas Kappeler, Riccardo Montalto Linear Algebra for the Sciences Thomas Kappeler, Riccardo Montalto 2 Contents Systems of linear equations 5. Linear systems with two equations and two unknowns............ 6.2 Gaussian elimination..............................

More information

MATH 304 Linear Algebra Lecture 24: Orthogonal complement. Orthogonal projection.

MATH 304 Linear Algebra Lecture 24: Orthogonal complement. Orthogonal projection. MATH 304 Linear Algebra Lecture 24: Orthogonal complement. Orthogonal projection. Euclidean structure Euclidean structure in R n includes: length of a vector: x, angle between vectors: θ, dot product:

More information

Math 1B03/1ZC3 - Tutorial 9. Mar. 14th/18th, 2014

Math 1B03/1ZC3 - Tutorial 9. Mar. 14th/18th, 2014 Math 1B03/1ZC3 - Tutorial 9 Mar. 14th/18th, 2014 Tutorial Info: Website: http://ms.mcmaster.ca/ dedieula. Math Help Centre: Wednesdays 2:30-5:30pm. Email: dedieula@math.mcmaster.ca. 1. Find a unit vector

More information

Cross Product or Vector Product

Cross Product or Vector Product Cross Product or Vector Product 1 Cross Product 1.1 Definitions Unlike the dot product, the cross product is only defined for 3-D vectors. In this section, when we use the word vector, we will mean 3-D

More information

Lesson 83. IBHL Lesson 84. The question arises: 1/16/16. So far, we have added two vectors and multiplied a vector by a scalar.

Lesson 83. IBHL Lesson 84. The question arises: 1/16/16. So far, we have added two vectors and multiplied a vector by a scalar. IBHL Lesson 84 VECTORS So far, we have added two vectors and multiplied a vector by a scalar. VECTORS The question arises: Is it possible to multiply two vectors so that their product is a useful quantity?

More information

Linear Algebra Test 2 Review by JC McNamara

Linear Algebra Test 2 Review by JC McNamara Linear Algebra Test 2 Review by JC McNamara 2.3 Properties of determinants: det(a T ) = det(a) det(ka) = k n det(a) det(a + B) det(a) + det(b) (In some cases this is true but not always) A is invertible

More information

INNER PRODUCTS. 1. Real Inner Products Definition 1. An inner product on a real vector space V is a function (u, v) u, v from V V to R satisfying

INNER PRODUCTS. 1. Real Inner Products Definition 1. An inner product on a real vector space V is a function (u, v) u, v from V V to R satisfying INNER PRODUCTS 1. Real Inner Products Definition 1. An inner product on a real vector space V is a function (u, v) u, v from V V to R satisfying (1) αu + βv, w = α u, w + β v, s for all u, v, w V and all

More information

Chapter 3 Linear Algebra Supplement

Chapter 3 Linear Algebra Supplement Chapter 3 Linear Algebra Supplement Properties of the Dot Product Note: this and the following two sections should logically appear at the end of these notes, but I have moved them to the front because

More information

Week 1 Vectors and Matrices

Week 1 Vectors and Matrices Week Vectors and Matrices Richard Earl Mathematical Institute, Oxford, OX 2LB, October 2003 Abstract Algebra and geometry of vectors The algebra of matrices 2x2 matrices Inverses Determinants Simultaneous

More information

Chapter 6. The Vector Product. 6.1 Parallel vectors

Chapter 6. The Vector Product. 6.1 Parallel vectors Chapter 6 The Vector Product 6 Parallel vectors Suppose that u and v are nonzero vectors We say that u and v are parallel, and write u v, if u is a scalar multiple of v (which will also force v to be a

More information

CHAPTER 10 VECTORS POINTS TO REMEMBER

CHAPTER 10 VECTORS POINTS TO REMEMBER CHAPTER 10 VECTORS POINTS TO REMEMBER A quantity that has magnitude as well as direction is called a vector It is denoted by a directed line segment Two or more vectors which are parallel to same line

More information

MA 52 May 9, Final Review

MA 52 May 9, Final Review MA 5 May 9, 6 Final Review This packet contains review problems for the whole course, including all the problems from the previous reviews. We also suggest below problems from the textbook for chapters

More information

10.4 Multiplying Vectors: The Cross Product

10.4 Multiplying Vectors: The Cross Product Arkansas Tech University MATH 2934: Calculus III Dr. Marcel B Finan 10.4 Multiplying Vectors: The Cross Product In this section we discuss another way of multiplying two vectors to obtain a vector, the

More information

6 Inner Product Spaces

6 Inner Product Spaces Lectures 16,17,18 6 Inner Product Spaces 6.1 Basic Definition Parallelogram law, the ability to measure angle between two vectors and in particular, the concept of perpendicularity make the euclidean space

More information

DOT PRODUCT AND CROSS PRODUCT. John P. D Angelo Dept. of Mathematics, Univ. of Illinois, 1409 W. Green St., Urbana IL

DOT PRODUCT AND CROSS PRODUCT. John P. D Angelo Dept. of Mathematics, Univ. of Illinois, 1409 W. Green St., Urbana IL DOT PRODUCT AND CROSS PRODUCT John P. D Angelo Dept. of Mathematics, Univ. of Illinois, 1409 W. Green St., Urbana IL 61801 jpda@math.uiuc.edu The algebra of vectors What is a vector? In physics a vector

More information

Outline. 1 Overview. 2 Area. 3 Introducing Arithmetic. 4 Interlude on Circles. Geometry, the Common Core, and Proof. John T. Baldwin, Andreas Mueller

Outline. 1 Overview. 2 Area. 3 Introducing Arithmetic. 4 Interlude on Circles. Geometry, the Common Core, and Proof. John T. Baldwin, Andreas Mueller November 28, 2012 Outline 1 2 3 4 5 Agenda 1 G-SRT4 Context. orems about similarity 2 Parallelograms - napoleon s theorem 3 : informally and formallly 4 s of parallelograms and triangles 5 a bit more on

More information

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

More information

3. Find the included angle between two nonzero vectors. (11,15,19,21,23,25,27)

3. Find the included angle between two nonzero vectors. (11,15,19,21,23,25,27) Section.: The Dot Product Objectives. Compute the dot product algebraically and geometrically. (,5,7,9). State 5 properties of the dot product.. Find the included angle between two nonzero vectors. (,5,9,,,5,7)

More information