ex) What is the component form of the vector shown in the picture above?

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "ex) What is the component form of the vector shown in the picture above?"

Transcription

1 Vectors A ector is a directed line segment, which has both a magnitude (length) and direction. A ector can be created using any two points in the plane, the direction of the ector is usually denoted by the placement of an arrow at the end of the line segment. ex) Pictured here is a ector (named) which has its initial point ( tail point) at ( 2, 1) and its terminal point (the arrow head ) at (2, 3). The component form of a ector is written using the form = ai+ bj. The alue of a is always written as a coefficient of i... which represents the ector s horizontal component. The alue of b is always written as a coefficient of j... which represents the ector s ertical component. ex) What is the component form of the ector shown in the picture aboe? The magnitude of a ector (which is denoted as ) is simply its length. It is calculated by applying the Pythagorean Theorem to the component coefficients a and b. 2 2 For any ector = ai+ bj, its magnitude is = a + b ex) What is the magnitude of the ector from the picture aboe? (Use the component form)

2 ex) Find the component form and the magnitude of the ector with initial point at ( 3, 11) and terminal point at(9, 40). (Approximate the magnitude to 2 decimal places.) The two main operations with ectors are ector addition and scalar multiplication. These operations can be done algebraically and graphically. Ex) For the ectors u= i+ 3j and = 2i j Plot ectors with their initial points at the origin and determine the following ector combinations. (a) u+ Find the sum algebraically (using component forms) Find the sum graphically (using parallelogram law) Calculate u+

3 (b) 2u+ 3 Find the sum algebraically (using component forms) Find the sum graphically (using parallelogram law) The 2 applied to u and the number 3 applied to are examples of scalar multiplication. The scalars will scale the length of each ector making them longer. Calculate 2u+ 3 (c) u Find the sum algebraically (using component forms) Find the sum graphically (using parallelogram law) Calculate u

4 Direction Angle for a Vector The direction angle is always considered to be the standard position angle starting on the positie x axis rotating counterclockwise to the ector s position. It can be determined by the formula: horizontal component tanθ = ertical component 1 To get the angle you ll need to use TAN on your calculator... BUT MAKE SURE YOUR ANGLE IS LOCATED IN THE CORRECT QUADRANT! ex) Determine the direction angle for the ector = 5i+ j. (IT ALWAYS HELPS TO SKETCH THE VECTOR FIRST) ex) Determine the direction angle for the ector = 4i+ 7j. 1 (SKETCH THE VECTOR FIRST AND BE CAREFUL USING TAN )

5 Unit Vectors When you want to presere the direction of a certain ector but apply a different length to it, you ll need to transform that ector in to a unit ector... essentially a ector of length 1. To get a unit ector, you diide the components by the magnitude: unit ector = ex) What is the unit ector which has the same direction as = 4i 3j? ex) A force of 1200 lbs is applied in the direction of the ector = 4i 3j. What are the components of this force ector? (Call the force ector F)

6 Decomposing a Vector When you already know the magnitude and direction angle, θ, of a ector, you can write it in component form using the formulas Horizontal Component cosθ Vertical Component sinθ Creating the ector = ( cos θ) i+ ( sin θ) j ex) Find the horizontal and ertical components of the ector with a length of = 800 and a direction ofθ = 145. (Round components to 2 decimal places). ex) A jet is flying in a direction of N 20 E with a speed of 500 mi/h. Represent the elocity of this jet as a ector in component form. (2 decimal place rounding)

7 Resultant Vectors The resultant of two or more ectors is the result of all of the ectors acting on the same object at the same time. A resultant is simply their ector sum. ex) Two tugboats are pulling a barge due north through a channel. One tugboat is pulling with a force of 3500 lbs at a heading of N 20 E and the other tugboat is pulling with 4000 lbs of force at a heading of N 25 W. (See the diagram.) a) What are the component forms of the force ector for each tugboat? b) Calculate the resultant ector. This is the ector sum of adding the tugboat ectors together. c) What is the magnitude and the direction of the resultant ector? Gie the direction as a bearing.

Section 9.1 Vectors in Two Dimensions

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

6.6 Vectors. Section 6.6 Notes Page 1

6.6 Vectors. Section 6.6 Notes Page 1 66 Vectors Section 66 Notes Page Vectors are needed in physics and engineering courses A ector is a quantity that has magnitude (size) in a certain direction You indicate a ector by a ray The length of

More information

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

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

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 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

Vectors. Vector Multiplication

Vectors. Vector Multiplication Vectors Directed Line Segments and Geometric Vectors A line segment to which a direction has been assigned is called a directed line segment. The figure below shows a directed line segment form P to Q.

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

Section V.2: Magnitudes, Directions, and Components of Vectors

Section V.2: Magnitudes, Directions, and Components of Vectors Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions

More information

A Review of Vector Addition

A Review of Vector Addition Motion and Forces in Two Dimensions Sec. 7.1 Forces in Two Dimensions 1. A Review of Vector Addition. Forces on an Inclined Plane 3. How to find an Equilibrant Vector 4. Projectile Motion Objectives Determine

More information

Basic Linear Algebra

Basic Linear Algebra Basic Linear Algebra by: Dan Sunday, softsurfer.com Table of Contents Coordinate Systems 1 Points and Vectors Basic Definitions Vector Addition Scalar Multiplication 3 Affine Addition 3 Vector Length 4

More information

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

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

Module 8 Lesson 4: Applications of Vectors

Module 8 Lesson 4: Applications of Vectors Module 8 Lesson 4: Applications of Vectors So now that you have learned the basic skills necessary to understand and operate with vectors, in this lesson, we will look at how to solve real world problems

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

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 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

PHYSICS 151 Notes for Online Lecture #6

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

Section 10.4 Vectors

Section 10.4 Vectors Section 10.4 Vectors A vector is represented by using a ray, or arrow, that starts at an initial point and ends at a terminal point. Your textbook will always use a bold letter to indicate a vector (such

More information

Review A: Vector Analysis

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

θ = 45 θ = 135 θ = 225 θ = 675 θ = 45 θ = 135 θ = 225 θ = 675 Trigonometry (A): Trigonometry Ratios You will learn:

θ = 45 θ = 135 θ = 225 θ = 675 θ = 45 θ = 135 θ = 225 θ = 675 Trigonometry (A): Trigonometry Ratios You will learn: Trigonometr (A): Trigonometr Ratios You will learn: () Concept of Basic Angles () how to form simple trigonometr ratios in all 4 quadrants () how to find the eact values of trigonometr ratios for special

More information

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

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

Trigonometry LESSON TWO - The Unit Circle Lesson Notes

Trigonometry LESSON TWO - The Unit Circle Lesson Notes (cosθ, sinθ) Trigonometry Example 1 Introduction to Circle Equations. a) A circle centered at the origin can be represented by the relation x 2 + y 2 = r 2, where r is the radius of the circle. Draw each

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

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

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

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

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

Unit 11 Additional Topics in Trigonometry - Classwork

Unit 11 Additional Topics in Trigonometry - Classwork Unit 11 Additional Topics in Trigonometry - Classwork In geometry and physics, concepts such as temperature, mass, time, length, area, and volume can be quantified with a single real number. These are

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

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

Vector Algebra. Addition: (A + B) + C = A + (B + C) (associative) Subtraction: A B = A + (-B)

Vector Algebra. Addition: (A + B) + C = A + (B + C) (associative) Subtraction: A B = A + (-B) Vector Algebra When dealing with scalars, the usual math operations (+, -, ) are sufficient to obtain any information needed. When dealing with ectors, the magnitudes can be operated on as scalars, but

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

Chapter 10: Terminology and Measurement in Biomechanics

Chapter 10: Terminology and Measurement in Biomechanics Chapter 10: Terminology and Measurement in Biomechanics KINESIOLOGY Scientific Basis of Human Motion, 11th edition Hamilton, Weimar & Luttgens Presentation Created by TK Koesterer, Ph.D., ATC Humboldt

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use a calculator to give the value to the nearest degree. 1) = sin-1(-0.4848) 1) -31 B)

More information

Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

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

Section 4.4 Velocity as a Vector

Section 4.4 Velocity as a Vector Section 4.4 Velocity as a Vector In elementary problems, the speed of a moing object is calculated by diiding the distance traelled by the trael time. In adanced work, speed is defined more efully as the

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

0017- Understanding and Using Vector and Transformational Geometries

0017- Understanding and Using Vector and Transformational Geometries Tom Coleman INTD 301 Final Project Dr. Johannes Vector geometry: 0017- Understanding and Using Vector and Transformational Geometries 3-D Cartesian coordinate representation: - A vector v is written as

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

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

Plotting and Adjusting Your Course: Using Vectors and Trigonometry in Navigation

Plotting and Adjusting Your Course: Using Vectors and Trigonometry in Navigation Plotting and Adjusting Your Course: Using Vectors and Trigonometry in Navigation ED 5661 Mathematics & Navigation Teacher Institute August 2011 By Serena Gay Target: Precalculus (grades 11 or 12) Lesson

More information

sin(θ) = opp hyp cos(θ) = adj hyp tan(θ) = opp adj

sin(θ) = opp hyp cos(θ) = adj hyp tan(θ) = opp adj Math, Trigonometr and Vectors Geometr 33º What is the angle equal to? a) α = 7 b) α = 57 c) α = 33 d) α = 90 e) α cannot be determined α Trig Definitions Here's a familiar image. To make predictive models

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

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

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

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

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

Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus

Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Trigonometry Review with the Unit Circle: All the trig. you ll ever need to know in Calculus Objectives: This is your review of trigonometry: angles, six trig. functions, identities and formulas, graphs:

More information

ELEMENTS OF VECTOR ALGEBRA

ELEMENTS OF VECTOR ALGEBRA 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

More information

Math, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image.

Math, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image. Math, Trigonometr and Vectors Geometr Trig Definitions Here's a familiar image. To make predictive models of the phsical world, we'll need to make visualizations, which we can then turn into analtical

More information

MATH 304 Linear Algebra Lecture 24: Scalar product.

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

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

Example 1. Example 1 Plot the points whose polar coordinates are given by

Example 1. Example 1 Plot the points whose polar coordinates are given by Polar Co-ordinates A polar coordinate system, gives the co-ordinates of a point with reference to a point O and a half line or ray starting at the point O. We will look at polar coordinates for points

More information

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

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

Chapter 6. Linear Transformation. 6.1 Intro. to Linear Transformation

Chapter 6. Linear Transformation. 6.1 Intro. to Linear Transformation Chapter 6 Linear Transformation 6 Intro to Linear Transformation Homework: Textbook, 6 Ex, 5, 9,, 5,, 7, 9,5, 55, 57, 6(a,b), 6; page 7- In this section, we discuss linear transformations 89 9 CHAPTER

More information

Introduction to Vectors

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

Vectors, velocity and displacement

Vectors, velocity and displacement Vectors, elocit and displacement Sample Modelling Actiities with Excel and Modellus ITforUS (Information Technolog for Understanding Science) 2007 IT for US - The project is funded with support from the

More information

Trigonometry (Chapter 6) Sample Test #1 First, a couple of things to help out:

Trigonometry (Chapter 6) Sample Test #1 First, a couple of things to help out: First, a couple of things to help out: Page 1 of 20 More Formulas (memorize these): Law of Sines: sin sin sin Law of Cosines: 2 cos 2 cos 2 cos Area of a Triangle: 1 2 sin 1 2 sin 1 2 sin 1 2 Solve the

More information

PHYS2001 Recitation 1 Friday, September 1, 2006

PHYS2001 Recitation 1 Friday, September 1, 2006 PHYS2001 Recitation 1 Friday, September 1, 2006 1. Consider the vectors A and B shown in the figure below. Which of the other four vectors in the figure (C, D, E, and F) best represents the direction of

More information

Solutions to Exercises, Section 6.1

Solutions to Exercises, Section 6.1 Instructor s Solutions Manual, Section 6.1 Exercise 1 Solutions to Exercises, Section 6.1 1. Find the area of a triangle that has sides of length 3 and 4, with an angle of 37 between those sides. 3 4 sin

More information

8-2 Vectors in the Coordinate Plane

8-2 Vectors in the Coordinate Plane Find the component form and magnitude of with the given initial and terminal points 1 A( 3, 1), B(4, 5) 3 A(10, 2), B(3, 5) First, find the component form First, find the component form Next, find the

More information

1.3 Displacement in Two Dimensions

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

Area and Arc Length in Polar Coordinates

Area and Arc Length in Polar Coordinates Area and Arc Length in Polar Coordinates The Cartesian Coordinate System (rectangular coordinates) is not always the most convenient way to describe points, or relations in the plane. There are certainly

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. 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

8-1 Introduction to Vectors

8-1 Introduction to Vectors State whether each quantity described is a vector quantity or a scalar quantity. 1. a box being pushed at a force of 125 newtons This quantity has a magnitude of 125 newtons, but no direction is given.

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

The Force Table Introduction: Theory:

The Force Table Introduction: Theory: 1 The Force Table Introduction: "The Force Table" is a simple tool for demonstrating Newton s First Law and the vector nature of forces. This tool is based on the principle of equilibrium. An object is

More information

Lecture 6. Weight. Tension. Normal Force. Static Friction. Cutnell+Johnson: 4.8-4.12, second half of section 4.7

Lecture 6. Weight. Tension. Normal Force. Static Friction. Cutnell+Johnson: 4.8-4.12, second half of section 4.7 Lecture 6 Weight Tension Normal Force Static Friction Cutnell+Johnson: 4.8-4.12, second half of section 4.7 In this lecture, I m going to discuss four different kinds of forces: weight, tension, the normal

More 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

Angles and Their Measure

Angles and Their Measure Trigonometry Lecture Notes Section 5.1 Angles and Their Measure Definitions: A Ray is part of a line that has only one end point and extends forever in the opposite direction. An Angle is formed by two

More information

Vectors and Scalars. AP Physics B

Vectors and Scalars. AP Physics B Vectors and Scalars P Physics Scalar SCLR is NY quantity in physics that has MGNITUDE, but NOT a direction associated with it. Magnitude numerical value with units. Scalar Example Speed Distance ge Magnitude

More information

UNITS, PHYSICAL QUANTITIES AND VECTORS

UNITS, PHYSICAL QUANTITIES AND VECTORS UNITS, PHYSICAL QUANTITIES AND VECTORS 1 1.37. IDENTIFY: Vector addition problem. We are given the magnitude and direction of three vectors and are asked to find their sum. SET UP: A B C 3.25 km 2.90 km

More information

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

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

Trigonometry Notes Sarah Brewer Alabama School of Math and Science. Last Updated: 25 November 2011

Trigonometry Notes Sarah Brewer Alabama School of Math and Science. Last Updated: 25 November 2011 Trigonometry Notes Sarah Brewer Alabama School of Math and Science Last Updated: 25 November 2011 6 Basic Trig Functions Defined as ratios of sides of a right triangle in relation to one of the acute angles

More information

AP Physics - Vector Algrebra Tutorial

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

1.7 Cylindrical and Spherical Coordinates

1.7 Cylindrical and Spherical Coordinates 56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the

More information

CHAPTER ONE VECTOR GEOMETRY

CHAPTER ONE VECTOR GEOMETRY CHAPTER ONE VECTOR GEOMETRY. INTRODUCTION In this chapter ectors are first introdced as geometric objects, namely as directed line segments, or arrows. The operations of addition, sbtraction, and mltiplication

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 , 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

Section 9.4 Trigonometric Functions of any Angle

Section 9.4 Trigonometric Functions of any Angle Section 9. Trigonometric Functions of any Angle So far we have only really looked at trigonometric functions of acute (less than 90º) angles. We would like to be able to find the trigonometric functions

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

FRICTION, WORK, AND THE INCLINED PLANE

FRICTION, WORK, AND THE INCLINED PLANE FRICTION, WORK, AND THE INCLINED PLANE Objective: To measure the coefficient of static and inetic friction between a bloc and an inclined plane and to examine the relationship between the plane s angle

More information

Addition and Resolution of Vectors Equilibrium of a Particle

Addition and Resolution of Vectors Equilibrium of a Particle Overview Addition and Resolution of Vectors Equilibrium of a Particle When a set of forces act on an object in such a way that the lines of action of the forces pass through a common point, the forces

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

Equilibrium of Forces Acting at a Point

Equilibrium of Forces Acting at a Point Equilibrium of Forces Acting at a Point Click here 1 for the revised instructions for this lab. INTRODUCTION Addition of Forces Forces are one of a group of quantities known as vectors, which are distinguished

More information

Addition and Subtraction of Vectors

Addition and Subtraction of Vectors ddition and Subtraction of Vectors 1 ppendi ddition and Subtraction of Vectors In this appendi the basic elements of vector algebra are eplored. Vectors are treated as geometric entities represented b

More information

Inverse Trigonometric Functions

Inverse Trigonometric Functions SECTION 4.7 Inverse Trigonometric Functions Copyright Cengage Learning. All rights reserved. Learning Objectives 1 2 3 4 Find the exact value of an inverse trigonometric function. Use a calculator to approximate

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. 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

Use Geometry Expressions to find equations of curves. Use Geometry Expressions to translate and dilate figures.

Use Geometry Expressions to find equations of curves. Use Geometry Expressions to translate and dilate figures. Learning Objectives Loci and Conics Lesson 2: The Circle Level: Precalculus Time required: 90 minutes Students are now acquainted with the idea of locus, and how Geometry Expressions can be used to explore

More information

PHYSICS 149: Lecture 4

PHYSICS 149: Lecture 4 PHYSICS 149: Lecture 4 Chapter 2 2.3 Inertia and Equilibrium: Newton s First Law of Motion 2.4 Vector Addition Using Components 2.5 Newton s Third Law 1 Net Force The net force is the vector sum of all

More information

Equilibrium of Concurrent Forces (Force Table)

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

(1.) The air speed of an airplane is 380 km/hr at a bearing of. Find the ground speed of the airplane as well as its

(1.) The air speed of an airplane is 380 km/hr at a bearing of. Find the ground speed of the airplane as well as its (1.) The air speed of an airplane is 380 km/hr at a bearing of 78 o. The speed of the wind is 20 km/hr heading due south. Find the ground speed of the airplane as well as its direction. Here is the diagram:

More information

Chapter 18 Static Equilibrium

Chapter 18 Static Equilibrium Chapter 8 Static Equilibrium 8. Introduction Static Equilibrium... 8. Lever Law... Example 8. Lever Law... 4 8.3 Generalized Lever Law... 5 8.4 Worked Examples... 7 Example 8. Suspended Rod... 7 Example

More information

2) When labeling bends that will be rotated, refer to the amount of rotation as the horizontal and/or vertical deflection angle.

2) When labeling bends that will be rotated, refer to the amount of rotation as the horizontal and/or vertical deflection angle. 13. Rotation of Fittings. a. Rotation of Bends. 1) Bends can be rotated about the pipe axis to produce a simultaneous deflection or combined bend. This section covers the design of horizontal bends rotated

More information

Class Notes for MATH 2 Precalculus. Fall Prepared by. Stephanie Sorenson

Class Notes for MATH 2 Precalculus. Fall Prepared by. Stephanie Sorenson Class Notes for MATH 2 Precalculus Fall 2012 Prepared by Stephanie Sorenson Table of Contents 1.2 Graphs of Equations... 1 1.4 Functions... 9 1.5 Analyzing Graphs of Functions... 14 1.6 A Library of Parent

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

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

More information

ANALYTICAL METHODS FOR ENGINEERS

ANALYTICAL METHODS FOR ENGINEERS UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

More information

Chapter 3B - Vectors. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University

Chapter 3B - Vectors. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University Chapter 3B - Vectors A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University 2007 Vectors Surveyors use accurate measures of magnitudes and directions to

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

The Dot Product. If v = a 1 i + b 1 j and w = a 2 i + b 2 j are vectors then their dot product is given by: v w = a 1 a 2 + b 1 b 2

The Dot Product. If v = a 1 i + b 1 j and w = a 2 i + b 2 j are vectors then their dot product is given by: v w = a 1 a 2 + b 1 b 2 The Dot Product In this section, e ill no concentrate on the vector operation called the dot product. The dot product of to vectors ill produce a scalar instead of a vector as in the other operations that

More information

Worksheet to Review Vector and Scalar Properties

Worksheet to Review Vector and Scalar Properties Worksheet to Review Vector and Scalar Properties 1. Differentiate between vectors and scalar quantities 2. Know what is being requested when the question asks for the magnitude of a quantity 3. Define

More information

3. KINEMATICS IN TWO DIMENSIONS; VECTORS.

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

Chapter 1 Quadratic Equations in One Unknown (I)

Chapter 1 Quadratic Equations in One Unknown (I) Tin Ka Ping Secondary School 015-016 F. Mathematics Compulsory Part Teaching Syllabus Chapter 1 Quadratic in One Unknown (I) 1 1.1 Real Number System A Integers B nal Numbers C Irrational Numbers D Real

More information

Introduction and Mathematical Concepts

Introduction and Mathematical Concepts CHAPTER 1 Introduction and Mathematical Concepts PREVIEW In this chapter you will be introduced to the physical units most frequently encountered in physics. After completion of the chapter you will be

More information