# ELEMENTS OF VECTOR ALGEBRA

Save this PDF as:

Size: px
Start display at page:

Download "ELEMENTS OF VECTOR ALGEBRA"

## Transcription

1 ELEMENTS OF VECTOR ALGEBRA A.1. VECTORS AND SCALAR QUANTITIES We have now proposed sets of basic dimensions and secondary dimensions to describe certain aspects of nature, but more than just dimensions identification and the number of units are often needed to convey adequately the desired information. For instance, to specify fully the velocity of a particle we must give: a. The magnitude of the secondary dimensions, by stating the number of scale units (m/s or km/h). b. The direction of the motion relative to some convenient reference. We have thus far considered only part a. To represent b we use an arrow to indicate direction and relate its length to the magnitude of the quantity involved. Certain quantities having magnitude and direction combine their effects in a special way. Thus the combined effect of two forces acting on a particle, as shown in Fig. A.1, corresponds to the diagonal of a parallelogram formed by the graphical representation of the forces. That is, the quantities add according to the parallelogram law. All quantities that have magnitude and direction and that add according to the parallelogram law are called vector quantities. Other quantities that have only magnitude, such as temperature and work, are called scalar quantities. A vector quantity will be denoted with a boldface letter, which in case of force becomes F. The student may ask: don t all quantities having magnitude and direction combine according to the parallelogram law and therefore become vector quantities? No, not all of them do. One very important example will be pointed out after we consider angular rotation. In the construction of the parallelogram it matters not which force is laid out first. In other words, F 1 combined with F 2 gives the same result as F 2 combined with F 1. In short, the combination is commutative. If the combination is not commutative, then it cannot in general be represented by a parallelogram operation and is thus not a vector. With this in mind, consider the angle of rotation of a body about some axis. We can associate

2 a magnitude (degrees or radians) and a direction (the axis and a stipulation of clockwise or counterclockwise) with this quantity. However, the angle of rotation cannot be considered a vector because in general two rotations about different axes cannot be replaced by a single rotation consistent with the parallelogram law. A.2. MAGNITUDE AND MULTIPLICATION OF A VECTOR BY A SCALAR The magnitude of a quantity, in strict mathematical parlance, is always a positive number of units whose value corresponds to the numerical measure of the quantity. Thus the magnitude of a quantity of measure 50 units is +50 units. The mathematical symbol for indicating the magnitude of a quantity is a set of vertical lines enclosing the quantity. Hence, the above verbal statement can be given mathematically as: -50 units = + 50 units Similarly, the magnitude of a vector quantity is a positive number of units corresponding to the length of the vector in those units. Using our vector symbols we can represent this as: magnitude of a vector A = A Thus A is a scalar quantity. We may now discuss the multiplication of a vector by a scalar. The definition of the product of A by scalar m, written simply as ma, is given as follows: ma is a vector having the same direction as A and the magnitude equal to the ordinary scalar product between magnitudes of m and A. If m is negative, it means simply that vector ma has a direction directly opposite to that of A. The vector A may be considered as the product of the scalar 1 and the vector A. Thus from the above statement we see that A differs from A in that it has an opposite sense. Furthermore, these operations have nothing to do with the line of

3 action of a vector, so A and A may have different lines of action (Fig A.2). This will be the case of the couple to be studied later. A.3. ADDITION AND SUBTRACTION OF VECTORS In adding a number of vectors, we may repeatedly employ the parallelogram construction. We can do this graphically by scaling the lengths of the arrows according to the magnitudes of the vector quantities they represent. The length of final arrow can be interpreted in terms of its length by employing the chosen scale factor. As an example, consider the coplanar vectors A, B, C shown in Fig. A.3. The addition of the vectors can be accomplished in two ways. (to be presented at the lecture). Note that the final vector is identical for both procedures. Thus: A + (B + C) = (A + B) + C When the quantities involved in an algebraic operation can be grouped without restriction, the operation is said to be associative. Thus, the addition of vectors is both commutative, as explained earlier, and associative. We may also add vectors by laying off the vectors head to tail. If many vectors/forces are involved, a polygon will be formed, and the sum of the vectors is the final closing side of the polygon (Fig.A.4.) The process of subtraction is defined in the following manner: to subtract vector B from vector A we reverse the direction of B (i.e., multiply by 1) and then add this new vector to A (Fig.A.5). A.4. RESOLUTION OF VECTORS; SCALAR COMPONENTS The reverse action of addition is called resolution. Thus for a given vector C, we may find a pair of vectors in any two stipulated directions coplanar with C such that the two component vectors sum to the original vector. This can again be accomplished by graphical construction or by using simple helpful sketches and

4 then employing trigonometric relations. (an example will be presented at the lecture). It is also readily possible to find three components not in the same plane as C by the preceding argument. Consider the specification of three orthogonal directions for the resolution of C as is shown in Fig.A.6. It is clear that the vectors C 1, C 2 and C 3 add up to the vector C and are hence called the orthogonal component vectors. The direction of a vector C relative to an orthogonal reference is given by the cosines of the angles formed by the vector and the coordinate axes. These are called direction cosines and are denoted as: cos(c,x) = cos α = l, cos(c,y) = cos β = m, (A.1) cos(c,z) = cos γ = n where α, β, and γ are associated with the x, y, and z axes respectively. It becomes clear from trigonometric considerations that C x = C 1 = C cosα = C l C y = C 2 = C cosα = C m (A.2) C z = C 3 = C cosα = C n Note that the orthogonal scalar components C x, C y and C z may be negative depending on the direction cosines. Using the Pythagorean theorem we obtain: / 2 C = [ C + C + C ] (A.3) x y z Sometimes only one of the vector components described above is desired. Then just one direction is prescribed, as shown in Fig.A.7. Thus the magnitude of the vector component C n is C cosδ. It is obvious that the triangle formed by the vector and its projection is a right triangle. In establishing C n we speak, therefore, of dropping a perpendicular from C to n.

5 A.5. UNIT VECTORS In describing vectors it is sometimes convenient to express them as the product of a scalar times a vector. To do this we use the unit vector, which has a magnitude of unity and a certain prescribed direction. It has no dimensions. To develop such a unit vector in the direction of the vector C, we may formulate the unit vector a so that C a (unit vector in direction C) = (A.4) C We can then express the vector C in the following form: C = C a (A.5) Unit vectors that are of particular use are those directed along the coordinate axes of an orthogonal reference, where i, j, and k correspond to the x, y and z directions as shown in Fig.A.8. Now the vector C from the Fig.A.6 can be expressed as follows C = C x i + C y j + C z k (A.6) A.6. SCALAR OR DOT PRODUCT OF TWO VECTORS In elementary physics, work was defined as the product of the force component in the direction of a displacement times the displacement. In effect, two vectors, force and displacement, are employed to give a scalar, work. In other physical problems, vectors are associated in this same manner so as to result in a scalar quantity. A vector operation that represents such operations concisely is the scalar product (or dot product), which for vectors A and B (see Fig. A.9) is defined as: A B = A B cosα (A.7) where α is the smaller angle between the two vectors. From the definition, it is clear that the dot product is commutative, since the number A B cosα is independent of the order of multiplication of its terms. Thus:

6 A B = B A (A.8) Let us consider A (B + C). By definition, we may project the vector (B + C) onto the direction of A and then multiply the magnitudes. However, the of the sum of two vectors is the same as the sum of the projections of the vectors, which means that: A (B + C) = A B + A C (A.9) An operation on a sum of quantities, which is the same as the sum of the operations on the quantities, is called a distributive operation. Thus the dot product is distributive. If a vector is multiplied by itself to produce a dot product, it forms the square of a number in the following manner: A A = A A = A 2 (A.10) If we express the vectors A and B in Cartesian components when taking the dot product, we get: A B = (A x i + A y j + A z k) (B x i + B y j + B z k) = A x B x + A y B y + A z B z (A.11) Thus, we see that a scalar product of two vectors is the sum of the ordinary products of the respective components. The dot product may be of immediate use in expressing the component of a vector along a given direction as discussed in Sec.A.4. If you refer back to Fig.A.7, you will recall that the component of C along the direction n is given as: C n = C cosδ (Α.12) Now let us consider a unit vector n along the direction of line n. If we carry out the dot product of C and n according to our fundamental definition, the result is: C n = C n cosδ (A.13) But since n is unity, when we compare the preceding equations it is apparent that: C n = C n (A.14) Similarly it is clear that the following useful relations are valid: C x = C i C y = C j C z = C k (A.15)

7 A.7. CROSS PRODUCT OF TWO VECTORS There are other interactions between the vector quantities that represent physical phenomena which result in vector quantities. One such interaction is the moment of a force (to be studied in one of next chapters). To set up a convenient operation for these situations, the vector cross product has been established. For the two vectors (having possibly different dimensions) shown in Fig.A.10 as A and B, the operation is defined as: A B = C (A.16) where C has a magnitude that is given as: C = A B sinα (Α.17) and a direction normal to the plane of AB. The sense, furthermore, corresponds to the direction of advanced for a right-hand screw rotated about C as an axis in the direction from A to B that is, from the first stated vector to the second stated vector through the smaller angle between them. In this case, the screw would advance upward in rotating from A to B, whether the procedure is viewed from above or below the plane AB. The description of vector C is now completed, since the magnitude and direction are fully established. The line of action of C in not determined by the cross product, for it depends on the situation. For this product the commutative law breaks down. We can verify, by considering the definition of cross product, that (Α B) = (B A) (A.18) As an exercise, you can demonstrate that the distributive law is still valid for this new operation. Thus A (B + C) = (A B) + (A C) (A.19) Next, consider the cross product of unit vectors. Here the product of equal vectors is zero because α and, consequently, sinα are zero. The product of i j is unity in magnitude and must have direction parallel to the z axis. If the z axis has been erected in a sense consistent with the right-hand screw rule when rotating

8 from the x to the y direction, the reference is called a right-hand triad, and we can write i j = k In this text, we will use a right-hand triad as a reference (Fig.A.11). For ease in evaluating unit cross products for such references, a simple permutation scheme is helpful. In Fig.A.12 the unit vectors i, j, and k are indicated on a circle in a clockwise sequence. Any cross product of a pair of unit vectors results in a positive third unit vector if going from the first vector to the second vector involves a clockwise motion on this circle. Otherwise the vector is negative. Thus: k j = i, k i = j, etc. Next, it will be useful to carry out the cross product in terms of Cartesian components. Using the results of preceding discussions, we get: Α B = (A x i + A y j + A z k) (B x i + B y j + B z k) = (A y B z A z B y )i + (A z B x A x B z )j + (A x B y A y B x )k (A.20) Another method of carrying out this long computation is to evaluate the following determinant: A x B i x A B j y y A B k z z (A.21) Bibliography Beer F.P, Johnston E.R., Jr., Vector Mechanics for Engineers, McGraw-Hill Shames I.H., Engineering Mechanics Statics, Prentice-Hall, 1959

### 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

### Section 1.2. Angles and the Dot Product. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.2 Angles and the Dot Product Suppose x = (x 1, x 2 ) and y = (y 1, y 2 ) are two vectors in R 2, neither of which is the zero vector 0. Let α and

More information

### 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

### 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

### 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

### 2. Spin Chemistry and the Vector Model

2. Spin Chemistry and the Vector Model The story of magnetic resonance spectroscopy and intersystem crossing is essentially a choreography of the twisting motion which causes reorientation or rephasing

More information

### Solution: 2. Sketch the graph of 2 given the vectors and shown below.

7.4 Vectors, Operations, and the Dot Product Quantities such as area, volume, length, temperature, and speed have magnitude only and can be completely characterized by a single real number with a unit

More information

### Announcements. 2-D Vector Addition

Announcements 2-D Vector Addition Today s Objectives Understand the difference between scalars and vectors Resolve a 2-D vector into components Perform vector operations Class Activities Applications Scalar

More information

### REVIEW OVER VECTORS. A scalar is a quantity that is defined by its value only. This value can be positive, negative or zero Example.

REVIEW OVER VECTORS I. Scalars & Vectors: A scalar is a quantity that is defined by its value only. This value can be positive, negative or zero Example mass = 5 kg A vector is a quantity that can be described

More information

### Vector Algebra II: Scalar and Vector Products

Chapter 2 Vector Algebra II: Scalar and Vector Products We saw in the previous chapter how vector quantities may be added and subtracted. In this chapter we consider the products of vectors and define

More information

### MATH 304 Linear Algebra Lecture 24: Scalar product.

MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B A A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent

More information

### Introduction to Statics. .PDF Edition Version 1.0. Notebook

Introduction to Statics.PDF Edition Version 1.0 Notebook Helen Margaret Lester Plants Late Professor Emerita Wallace Starr Venable Emeritus Associate Professor West Virginia University, Morgantown, West

More information

### 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

### Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

More 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 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

### Vectors. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) Vectors Spring /

Vectors Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) Vectors Spring 2012 1 / 18 Introduction - Definition Many quantities we use in the sciences such as mass, volume, distance, can be expressed

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 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

### 2 Session Two - Complex Numbers and Vectors

PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar

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 a brick on a table Vector has a magnitude and a direction Scalar has a magnitude

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 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

### Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

More information

### 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

### Mechanics lecture 7 Moment of a force, torque, equilibrium of a body

G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and

More information

### Vectors What are Vectors? which measures how far the vector reaches in each direction, i.e. (x, y, z).

1 1. What are Vectors? A vector is a directed line segment. A vector can be described in two ways: Component form Magnitude and Direction which measures how far the vector reaches in each direction, i.e.

More information

### PHYSICS 151 Notes for Online Lecture #6

PHYSICS 151 Notes for Online Lecture #6 Vectors - A vector is basically an arrow. The length of the arrow represents the magnitude (value) and the arrow points in the direction. Many different quantities

More information

### 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

### Review A: Vector Analysis

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Review A: Vector Analysis A... A-0 A.1 Vectors A-2 A.1.1 Introduction A-2 A.1.2 Properties of a Vector A-2 A.1.3 Application of Vectors

More information

### Equilibrium of Concurrent Forces (Force Table)

Equilibrium of Concurrent Forces (Force Table) Objectives: Experimental objective Students will verify the conditions required (zero net force) for a system to be in equilibrium under the influence of

More information

### Vectors are quantities that have both a direction and a magnitude (size).

Scalars & Vectors Vectors are quantities that have both a direction and a magnitude (size). Ex. km, 30 ο north of east Examples of Vectors used in Physics Displacement Velocity Acceleration Force Scalars

More information

### 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

### 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

### Vectors Math 122 Calculus III D Joyce, Fall 2012

Vectors Math 122 Calculus III D Joyce, Fall 2012 Vectors in the plane R 2. A vector v can be interpreted as an arro in the plane R 2 ith a certain length and a certain direction. The same vector can be

More information

### Chapter 3 Vectors. m = m1 + m2 = 3 kg + 4 kg = 7 kg (3.1)

COROLLARY I. A body, acted on by two forces simultaneously, will describe the diagonal of a parallelogram in the same time as it would describe the sides by those forces separately. Isaac Newton - Principia

More information

### Lecture PowerPoints. Chapter 3 Physics: Principles with Applications, 6 th edition Giancoli

Lecture PowerPoints Chapter 3 Physics: Principles with Applications, 6 th edition Giancoli 2005 Pearson Prentice Hall This work is protected by United States copyright laws and is provided solely for the

More information

### Vector algebra Christian Miller CS Fall 2011

Vector algebra Christian Miller CS 354 - Fall 2011 Vector algebra A system commonly used to describe space Vectors, linear operators, tensors, etc. Used to build classical physics and the vast majority

More information

### 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

### 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

### 3. KINEMATICS IN TWO DIMENSIONS; VECTORS.

3. KINEMATICS IN TWO DIMENSIONS; VECTORS. Key words: Motion in Two Dimensions, Scalars, Vectors, Addition of Vectors by Graphical Methods, Tail to Tip Method, Parallelogram Method, Negative Vector, Vector

More information

### THEORETICAL MECHANICS

PROF. DR. ING. VASILE SZOLGA THEORETICAL MECHANICS LECTURE NOTES AND SAMPLE PROBLEMS PART ONE STATICS OF THE PARTICLE, OF THE RIGID BODY AND OF THE SYSTEMS OF BODIES KINEMATICS OF THE PARTICLE 2010 0 Contents

More information

### Modern Geometry Homework.

Modern Geometry Homework. 1. Rigid motions of the line. Let R be the real numbers. We define the distance between x, y R by where is the usual absolute value. distance between x and y = x y z = { z, z

More information

### EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS TUTORIAL 1 NON-CONCURRENT COPLANAR FORCE SYSTEMS 1. Be able to determine the effects

More information

### 1. Introduction identity algbriac factoring identities

1. Introduction An identity is an equality relationship between two mathematical expressions. For example, in basic algebra students are expected to master various algbriac factoring identities such as

More information

### Vectors, Gradient, Divergence and Curl.

Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use

More information

### Vector Definition. Chapter 1. Example 2 (Position) Example 1 (Position) Activity: What is the position of the center of your tabletop?

Vector Definition Chapter 1 Vectors A quantity that has two properties: magnitude and direction It is represented by an arrow; visually the length represents magnitude It is typically drawn on a coordinate

More information

### 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

### NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document

More information

### AP Physics - Vector Algrebra Tutorial

AP Physics - Vector Algrebra Tutorial Thomas Jefferson High School for Science and Technology AP Physics Team Summer 2013 1 CONTENTS CONTENTS Contents 1 Scalars and Vectors 3 2 Rectangular and Polar Form

More information

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

More information

### Introduction to Vectors

Introduction to Vectors A vector is a physical quantity that has both magnitude and direction. An example is a plane flying NE at 200 km/hr. This vector is written as 200 Km/hr at 45. Another example is

More information

### Kyu-Jung Kim Mechanical Engineering Department, California State Polytechnic University, Pomona, U.S.A.

MECHANICS: STATICS AND DYNAMICS Kyu-Jung Kim Mechanical Engineering Department, California State Polytechnic University, Pomona, U.S.A. Keywords: mechanics, statics, dynamics, equilibrium, kinematics,

More information

### PURE MATHEMATICS AM 27

AM Syllabus (015): Pure Mathematics AM SYLLABUS (015) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (015): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)

More information

### PURE MATHEMATICS AM 27

AM SYLLABUS (013) PURE MATHEMATICS AM 7 SYLLABUS 1 Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics and

More information

### VECTORS. A vector is a quantity that has both magnitude and direction.

VECTOS Definition: A vector is a quantity that has both magnitude and direction. NOTE: The position of a vector has no bearing on its definition. A vector can be slid horizontally or vertically without

More information

### Difference between a vector and a scalar quantity. N or 90 o. S or 270 o

Vectors Vectors and Scalars Distinguish between vector and scalar quantities, and give examples of each. method. A vector is represented in print by a bold italicized symbol, for example, F. A vector has

More information

### Section 12.1 Translations and Rotations

Section 12.1 Translations and Rotations Any rigid motion that preserves length or distance is an isometry (meaning equal measure ). In this section, we will investigate two types of isometries: translations

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,

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

### 1.3 Displacement in Two Dimensions

1.3 Displacement in Two Dimensions So far, you have learned about motion in one dimension. This is adequate for learning basic principles of kinematics, but it is not enough to describe the motions of

More information

### 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

### Mechanics 1: Vectors

Mechanics 1: Vectors roadly speaking, mechanical systems will be described by a combination of scalar and vector quantities. scalar is just a (real) number. For example, mass or weight is characterized

More information

### 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

### Linear Algebra: Vectors

A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector

More information

### 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

### Vectors-Algebra and Geometry

Chapter Two Vectors-Algebra and Geometry 21 Vectors A directed line segment in space is a line segment together with a direction Thus the directed line segment from the point P to the point Q is different

More 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

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

### 9.3. Direction Ratios and Direction Cosines. Introduction. Prerequisites. Learning Outcomes. Learning Style

Direction Ratios and Direction Cosines 9.3 Introduction Direction ratios provide a convenient way of specifying the direction of a line in three dimensional space. Direction cosines are the cosines of

More information

### 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

### Figuring out the amplitude of the sun an explanation of what the spreadsheet is doing.

Figuring out the amplitude of the sun an explanation of what the spreadsheet is doing. The amplitude of the sun (at sunrise, say) is the direction you look to see the sun come up. If it s rising exactly

More information

### 3-D Dynamics of Rigid Bodies

3-D Dynamics of Rigid Bodies Introduction of third dimension :: Third component of vectors representing force, linear velocity, linear acceleration, and linear momentum :: Two additional components for

More information

### Review of Vector Analysis in Cartesian Coordinates

R. evicky, CBE 6333 Review of Vector Analysis in Cartesian Coordinates Scalar: A quantity that has magnitude, but no direction. Examples are mass, temperature, pressure, time, distance, and real numbers.

More information

### Cartesian Coordinate System. Also called rectangular coordinate system x- and y- axes intersect at the origin Points are labeled (x,y)

Physics 1 Vectors Cartesian Coordinate System Also called rectangular coordinate system x- and y- axes intersect at the origin Points are labeled (x,y) Polar Coordinate System Origin and reference line

More information

### Section V.4: Cross Product

Section V.4: Cross Product Definition The cross product of vectors A and B is written as A B. The result of the cross product A B is a third vector which is perpendicular to both A and B. (Because the

More information

### Vector Spaces; the Space R n

Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which

More information

### Vectors 2. The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996.

Vectors 2 The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996. Launch Mathematica. Type

More information

### WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS

Visit this link to read the introductory text for this syllabus. 1. Circular Measure Lengths of Arcs of circles and Radians Perimeters of Sectors and Segments measure in radians 2. Trigonometry (i) Sine,

More information

### 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

### Section 9.1 Vectors in Two Dimensions

Section 9.1 Vectors in Two Dimensions Geometric Description of Vectors A vector in the plane is a line segment with an assigned direction. We sketch a vector as shown in the first Figure below with an

More information

### Lecture 4. Vectors. Motion and acceleration in two dimensions. Cutnell+Johnson: chapter ,

Lecture 4 Vectors Motion and acceleration in two dimensions Cutnell+Johnson: chapter 1.5-1.8, 3.1-3.3 We ve done motion in one dimension. Since the world usually has three dimensions, we re going to do

More information

### Chapter 2: Concurrent force systems. Department of Mechanical Engineering

Chapter : Concurrent force sstems Objectives To understand the basic characteristics of forces To understand the classification of force sstems To understand some force principles To know how to obtain

More information

### FORCE VECTORS. Lecture s Objectives

CHAPTER Engineering Mechanics: Statics FORCE VECTORS Tenth Edition College of Engineering Department of Mechanical Engineering 2b by Dr. Ibrahim A. Assakkaf SPRING 2007 ENES 110 Statics Department of Mechanical

More information

### Lab 2: Vector Analysis

Lab 2: Vector Analysis Objectives: to practice using graphical and analytical methods to add vectors in two dimensions Equipment: Meter stick Ruler Protractor Force table Ring Pulleys with attachments

More information

### Montana Common Core Standard

Algebra 2 Grade Level: 10(with Recommendation), 11, 12 Length: 1 Year Period(s) Per Day: 1 Credit: 1 Credit Requirement Fulfilled: Mathematics Course Description This course covers the main theories in

More information

### How is a vector rotated?

How is a vector rotated? V. Balakrishnan Department of Physics, Indian Institute of Technology, Madras 600 036 Appeared in Resonance, Vol. 4, No. 10, pp. 61-68 (1999) Introduction In an earlier series

More information

### 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

### Engineering Mechanics I. Phongsaen PITAKWATCHARA

2103-213 Engineering Mechanics I Phongsaen.P@chula.ac.th May 13, 2011 Contents Preface xiv 1 Introduction to Statics 1 1.1 Basic Concepts............................ 2 1.2 Scalars and Vectors..........................

More information

### Interactive Math Glossary Terms and Definitions

Terms and Definitions Absolute Value the magnitude of a number, or the distance from 0 on a real number line Additive Property of Area the process of finding an the area of a shape by totaling the areas

More information

### Introduction. The Aims & Objectives of the Mathematical Portion of the IBA Entry Test

Introduction The career world is competitive. The competition and the opportunities in the career world become a serious problem for students if they do not do well in Mathematics, because then they are

More information

### Trigonometry Lesson Objectives

Trigonometry Lesson Unit 1: RIGHT TRIANGLE TRIGONOMETRY Lengths of Sides Evaluate trigonometric expressions. Express trigonometric functions as ratios in terms of the sides of a right triangle. Use the

More information

### Overview Mathematical Practices Congruence

Overview Mathematical Practices Congruence 1. Make sense of problems and persevere in Experiment with transformations in the plane. solving them. Understand congruence in terms of rigid motions. 2. Reason

More information

### Section V.3: Dot Product

Section V.3: Dot Product Introduction So far we have looked at operations on a single vector. There are a number of ways to combine two vectors. Vector addition and subtraction will not be covered here,

More information

### Applications of Trigonometry

chapter 6 Tides on a Florida beach follow a periodic pattern modeled by trigonometric functions. Applications of Trigonometry This chapter focuses on applications of the trigonometry that was introduced

More information

### Rotation Matrices. Suppose that 2 R. We let

Suppose that R. We let Rotation Matrices R : R! R be the function defined as follows: Any vector in the plane can be written in polar coordinates as rcos, sin where r 0and R. For any such vector, we define

More information

### ectors and Application P(x, y, z)! \$ ! \$ & " 11,750 12,750 13,750

thstrack MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) odule 3 Topic 3 Module 9 Introduction Vectors and Applications to Matrices ectors

More information

### 6. LECTURE 6. Objectives

6. LECTURE 6 Objectives I understand how to use vectors to understand displacement. I can find the magnitude of a vector. I can sketch a vector. I can add and subtract vector. I can multiply a vector by

More information

### Mathematical Procedures

CHAPTER 6 Mathematical Procedures 168 CHAPTER 6 Mathematical Procedures The multidisciplinary approach to medicine has incorporated a wide variety of mathematical procedures from the fields of physics,

More information

### Maths for Computer Graphics

Analytic Geometry Review of geometry Euclid laid the foundations of geometry that have been taught in schools for centuries. In the last century, mathematicians such as Bernhard Riemann (1809 1900) and

More information

### Universal Law of Gravitation

Universal Law of Gravitation Law: Every body exerts a force of attraction on every other body. This force called, gravity, is relatively weak and decreases rapidly with the distance separating the bodies

More information

### 6. ISOMETRIES isometry central isometry translation Theorem 1: Proof:

6. ISOMETRIES 6.1. Isometries Fundamental to the theory of symmetry are the concepts of distance and angle. So we work within R n, considered as an inner-product space. This is the usual n- dimensional

More information

### Content. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11

Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter

More information

### In order to describe motion you need to describe the following properties.

Chapter 2 One Dimensional Kinematics How would you describe the following motion? Ex: random 1-D path speeding up and slowing down In order to describe motion you need to describe the following properties.

More information

### (Mathematics Syllabus Form 3 Track 3 for Secondary Schools June 2013) Page 1 of 9

(Mathematics Syllabus Form 3 Track 3 for Secondary Schools June 2013) Page 1 of 9 Contents Pages Number and Applications 3-4 Algebra 5-6 Shape Space and Measurement 7-8 Data Handling 9 (Mathematics Syllabus

More information