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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 Bellwork a.) Write the line 2x - 4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a c.) Find the slope of the line perpendicular to part b or a May 8 7:30 AM 1

2 Day 1 I. Vector Vocabulary A.) Scalar- This has magnitude but no direction. B.) Vectors- These have both magnitude and direction. Examples Forces, accelertion, momentum, weight and translations in geometry May 15 12:20 PM 2

3 C.) Equality 2 or more vectors are equal if they have the same magnitude and direction. D.) Zero Vector 0 = E.) Addition Let a = and b = then a + b = F.) Subtraction Let a = and b = then a - b = May 18 10:03 AM 3

4 G.) Scalar Multiplication Let a = and k is a scalar then ka = Examples 1.) Let a = b = Find a.) 5 a b.) -b + 3a Apr 20 1:30 PM 4

5 II.) Position vectors Position Vector- a vector with the additional property that it is fixed at its back end to the origin. B( b 1, b 2 ) A ( a1, a 2) AB = AO + OB = b - a So AB = May 18 10:03 AM 5

6 Example 1.) Given P( 2,5) and Q(3,-1) a.) write P and Q as a position vectors 2.) Given AB = and BC = find AC 3.) What if BA = and BC = find AC May 18 10:10 AM 6

7 III. Magnitude A.) Magnitude is the length or the magnitude of a vector is the absolute value. Formula **only 3 dimensional given on IB a = v = = remind you of distance formula? May 15 12:21 PM 7

8 Examples 1.) Find the length of the following vectors a.) b.) May 19 3:05 PM 8

9 try... 2.) Given A= ( 3,-2) and B=( 1,2) a.) OA b.) AB c.) AB Apr 20 1:52 PM 9

10 Bellwork Let a = 2 and b = Find a.) a b.) b c.) a + b d.) a + b e.) Show that a + b < a + b May 21 1:32 PM 10

11 Day 2 III. 3 Dimensional A.) Magnitude is the length or the magnitude of a vector is the absolute value. a = v = = yesterday a = v = Formula **only 3 dimensional given on IB May 15 12:21 PM 11

12 B.) Distance and midpoint 1.) recall 2 dimensional 2.) 3 dimensional AB = [AB] = ( x 2 + x 1, y 2 + y 1, z 1 + z 2 ) So USE FORMULA sheet Apr 26 9:48 AM 12

13 Examples 1.) Find the length of the following vectors a.) Think about what we did yesterday b.) If they don't ask in words, then they use May 19 3:05 PM 13

14 2.) Find the position vectors given A= ( 3,-2,5) and B=( 1,0,-2) a.) OA b.) AB c.) BA You try... d.) [AB] 3.) Given A= ( 2,2,4) and B =( -1,1,2) find a.) AB b.) AB c.) [AB] Apr 20 1:52 PM 14

15 Bellwork P= ( -1,2,4) and Q=( 0,2,5) a.) What are P and Q? Points or Vectors b.) Find PQ c.) Find PQ d.) Find [PQ] Apr 25 12:21 PM 15

16 Geometry Review Slope Formula Parallel lines y= 2x + 3 2x + y = 5 2x - y = 6 Perpendicular lines y = 3x - 1 3y = x + 2 3y = 4 - x Apr 4 7:22 AM 16

17 Day 3 IV. Parallel vectors A.) Vectors a and b are said to be parallel if there is a scalar number k such that a = kb. Examples 1.) Show that a = and b = are parallel. May 19 2:55 PM 17

18 2.) Find the values of a and b where the vectors below are parallel a b ) Let a = 2 and b = 3 1 Find the scalar factor k where a = k b. May 19 2:57 PM 18

19 Bellwork Let a = and b = Find x and y where a and b are parallel Apr 26 12:19 PM 19

20 Day 4 II. Unit Vectors A.) A Unit vector is a vector with a length of 1. ex: 2D i = 1 0 j= 0 1 recall distance formula or magnitude 3D i = j = k = component form B.) A zero vector is a vector quantity with no direction. ex 0 0 May 15 12:20 PM 20

21 C.) Finding Unit Vectors 1.) v = v 1 v 2 = v 1 i + v 2 j unit vector form a i + b j = ( ai + bj) = 1 v v = v v This vector will have length 1. Examples Find a unit vector in the same direction as 1.) 5 i - 2 j 2.) ) Find the value of x when x is a unit vector ¼ May 15 12:21 PM 21

22 4.) Finding a unit vector, b, in the same direction Formula k a a Example Find a unit vector, b, if the length is 5 in the same direction as Apr 26 10:07 AM 22

23 C.) Parallel unit vectors formula = ± k a a 1.) Find the unit vector that are parallel to a.) b.) c.) 6i - 5j May 15 12:22 PM 23

24 Bellwork Given the following vector find, in simplest radical form : a.) a unit vector b.) a unit vector traveling in the opposite direction c.) a vector traveling in the same direction as the given but having length of 6 May 13 3:23 PM 24

25 Bellwork find the length: u = 2i - 6j +2k May 28 8:43 AM 25

26 Day 6 VIII Products A.) Inner Product, scalar, or dot product Let v = and w = v w = v 1 w 1 + v 2 w 2 true also for 3 dimensions Let v = v 1 v 2 v 3 and w = w 1 w 2 w 3 * given on IB exam v w = v 1 w 1 + v 2 w 2 May 15 12:22 PM 26

27 Examples Find the inner product of a and b where 1.) a = ( 3,5) and b = ( 8,-3) 2.) a = ( 2,-1,3) and b = ( 5,3,0) May 21 1:37 PM 27

28 B.) Angles v w = v w cos θ This is used to find angles between vectors. rework the formula v w = v 1 w 1 + v 2 w 2 so v 1 w 1 + v 2 w 2 v w = cos θ or simply May 15 12:22 PM 28

29 Find the angle between the vectors v= -i + 3j and w = -i + 2j First Find the inner product Second Find the magnitude of each Third Plug into formula v 1 w 1 + v 2 w 2 v w = cos θ May 21 1:42 PM 29

30 v 1 w 1 + v 2 w 2 + v 3 w 3 v w = cos θ example # May 21 1:44 PM 30

31 bellwork What is the angle between the vectors and Apr 4 11:55 AM 31

32 Day 7 C.) Perpendicular Vectors If the inner product is equal to zero, then the vectors will be perpendicular. Why? graph y = cos x Examples: 1.) Determine if a and b are perpendicular vectors a = b= 2.) Determine if a and b are perpendicular vectors a= b= May 15 12:23 PM 32

33 3.) How about and? 4.) Find the value of x which will make the vectors perpendicular? 5.) Given the vertices of triangle KLM K( 4,-2,7), L( 6,1,-1), and M ( 3,2,-2), find a.) the position vectors KL and LM b.)measure of < KLM Jun 1 9:57 AM 33

34 6.) Given the vertices of ABC using scalar products, determine if it is a right triangle. A(-2,6) B(4,5) C(1,-4) May 16 8:57 AM 34

35 Bellwork Find the values of x which will make the 2 vectors perpendicular Apr 25 12:23 PM 35

36 Day 8 D.) Equations of lines slope = gradient direction vector = notice relationship with slope Examples 1.) Given two lines 2x - y = 6 and x + 3y = 4. Find the angle between these 2 lines. Apr 26 1:48 PM 36

37 2.) same question use this time x - y= 3 and 3x + 2y = 11 Apr 26 1:51 PM 37

38 Bellwork Two vectors are defined as a = 2i + xj and b = i - 4j Find the value of x if 1.) the vectors were parallel 2.) the vectors were perpendicular Jun 1 9:59 AM 38

39 bellwork p = q = 1.) Find p + q 2.) p - 1/2 q 3.) 4.) p + q May 21 9:30 AM 39

40 Bellwork Given: and Find the exact value of: May 15 3:00 PM 40

41 Bellwork Given the vertices of ABC using scalar products, determine if it is a right triangle. A(-2,6) B(4,5) C(1,-4) May 16 8:57 AM 41

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

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

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

Equations Involving Lines and Planes Standard equations for lines in space

Equations Involving Lines and Planes Standard equations for lines in space Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity

More 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

9 Multiplication of Vectors: The Scalar or Dot Product

9 Multiplication of Vectors: The Scalar or Dot Product Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation

More information

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

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

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

C relative to O being abc,, respectively, then b a c. 2 EP-Program - Strisuksa School - Roi-et Math : Vectors Dr.Wattana Toutip - Department of Mathematics Khon Kaen University 200 :Wattana Toutip wattou@kku.ac.th http://home.kku.ac.th/wattou 2. Vectors A

More information

CHAPTER FIVE. 5. Equations of Lines in R 3

CHAPTER FIVE. 5. Equations of Lines in R 3 118 CHAPTER FIVE 5. Equations of Lines in R 3 In this chapter it is going to be very important to distinguish clearly between points and vectors. Frequently in the past the distinction has only been a

More information

Co-ordinate Geometry THE EQUATION OF STRAIGHT LINES

Co-ordinate Geometry THE EQUATION OF STRAIGHT LINES Co-ordinate Geometry THE EQUATION OF STRAIGHT LINES This section refers to the properties of straight lines and curves using rules found by the use of cartesian co-ordinates. The Gradient of a Line. As

More information

Analytical Geometry (4)

Analytical Geometry (4) Analytical Geometry (4) Learning Outcomes and Assessment Standards Learning Outcome 3: Space, shape and measurement Assessment Standard As 3(c) and AS 3(a) The gradient and inclination of a straight line

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

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

Vector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A. 1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called

More information

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

Math 241, Exam 1 Information.

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

More information

FURTHER VECTORS (MEI)

FURTHER VECTORS (MEI) Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

More information

LINES AND PLANES CHRIS JOHNSON

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

More information

Section V.3: Dot Product

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

12.5 Equations of Lines and Planes

12.5 Equations of Lines and Planes Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P

More information

L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

More information

Section 9.5: Equations of Lines and Planes

Section 9.5: Equations of Lines and Planes Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 3-5 odd, 2-37 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that

More information

5-1 Perpendicular and Angle Bisectors

5-1 Perpendicular and Angle Bisectors 5-1 Perpendicular and Angle Bisectors Warm Up Lesson Presentation Lesson Quiz Geometry Warm Up Construct each of the following. 1. A perpendicular bisector. 2. An angle bisector. 3. Find the midpoint and

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

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

x(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3

x(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3 CORE 4 Summary Notes Rational Expressions Factorise all expressions where possible Cancel any factors common to the numerator and denominator x + 5x x(x + 5) x 5 (x + 5)(x 5) x x 5 To add or subtract -

More information

SQA Higher Mathematics Unit 3

SQA Higher Mathematics Unit 3 SCHOLAR Study Guide SQA Higher Mathematics Unit 3 Jane Paterson Heriot-Watt University Dorothy Watson Balerno High School Heriot-Watt University Edinburgh EH14 4AS, United Kingdom. First published 2001

More information

PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71. Applications. F = mc + b.

PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71. Applications. F = mc + b. PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71 Applications The formula y = mx + b sometimes appears with different symbols. For example, instead of x, we could use the letter C.

More information

Graphing - Parallel and Perpendicular Lines

Graphing - Parallel and Perpendicular Lines . Graphing - Parallel and Perpendicular Lines Objective: Identify the equation of a line given a parallel or perpendicular line. There is an interesting connection between the slope of lines that are parallel

More information

CIRCLE COORDINATE GEOMETRY

CIRCLE COORDINATE GEOMETRY CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle

More information

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

Name Class. Date Section. Test Form A Chapter 11. Chapter 11 Test Bank 155 Chapter Test Bank 55 Test Form A Chapter Name Class Date Section. Find a unit vector in the direction of v if v is the vector from P,, 3 to Q,, 0. (a) 3i 3j 3k (b) i j k 3 i 3 j 3 k 3 i 3 j 3 k. Calculate

More information

Mathematics 205 HWK 6 Solutions Section 13.3 p627. Note: Remember that boldface is being used here, rather than overhead arrows, to indicate vectors.

Mathematics 205 HWK 6 Solutions Section 13.3 p627. Note: Remember that boldface is being used here, rather than overhead arrows, to indicate vectors. Mathematics 205 HWK 6 Solutions Section 13.3 p627 Note: Remember that boldface is being used here, rather than overhead arrows, to indicate vectors. Problem 5, 13.3, p627. Given a = 2j + k or a = (0,2,

More information

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

EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS

EXPONENTS. To the applicant: KEY WORDS AND CONVERTING WORDS TO EQUATIONS To the applicant: The following information will help you review math that is included in the Paraprofessional written examination for the Conejo Valley Unified School District. The Education Code requires

More information

www.sakshieducation.com

www.sakshieducation.com LENGTH OF THE PERPENDICULAR FROM A POINT TO A STRAIGHT LINE AND DISTANCE BETWEEN TWO PAPALLEL LINES THEOREM The perpendicular distance from a point P(x 1, y 1 ) to the line ax + by + c 0 is ax1+ by1+ c

More information

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

Two vectors are equal if they have the same length and direction. They do not Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must

More information

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

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

Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE

Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE 1 P a g e Motion Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE If an object changes its position with respect to its surroundings with time, then it is called in motion. Rest If an object

More information

What are the place values to the left of the decimal point and their associated powers of ten?

What are the place values to the left of the decimal point and their associated powers of ten? The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

More information

Chapter 4.1 Parallel Lines and Planes

Chapter 4.1 Parallel Lines and Planes Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about

More information

Equation of a Line. Chapter H2. The Gradient of a Line. m AB = Exercise H2 1

Equation of a Line. Chapter H2. The Gradient of a Line. m AB = Exercise H2 1 Chapter H2 Equation of a Line The Gradient of a Line The gradient of a line is simpl a measure of how steep the line is. It is defined as follows :- gradient = vertical horizontal horizontal A B vertical

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

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 13.5 Equations of Lines and Planes

Section 13.5 Equations of Lines and Planes Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.

More information

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

Vectors Math 122 Calculus III D Joyce, Fall 2012

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

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

Lines. We have learned that the graph of a linear equation. y = mx +b

Lines. We have learned that the graph of a linear equation. y = mx +b Section 0. Lines We have learne that the graph of a linear equation = m +b is a nonvertical line with slope m an -intercept (0, b). We can also look at the angle that such a line makes with the -ais. This

More information

Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

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

GRADE 12 SEPTEMBER 2014 MATHEMATICS P2

GRADE 12 SEPTEMBER 2014 MATHEMATICS P2 NATIONAL SENIOR CERTIFICATE GRADE 1 SEPTEMBER 014 MATHEMATICS P MARKS: 150 TIME: 3 hours *MATHE* This question paper consists of 15 pages, including diagram sheets and 1 information sheet. MATHEMATICS

More information

Sect The Slope-Intercept Form

Sect The Slope-Intercept Form Concepts # and # Sect. - The Slope-Intercept Form Slope-Intercept Form of a line Recall the following definition from the beginning of the chapter: Let a, b, and c be real numbers where a and b are not

More information

Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

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

Section 1: How will you be tested? This section will give you information about the different types of examination papers that are available.

Section 1: How will you be tested? This section will give you information about the different types of examination papers that are available. REVISION CHECKLIST for IGCSE Mathematics 0580 A guide for students How to use this guide This guide describes what topics and skills you need to know for your IGCSE Mathematics examination. It will help

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

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m.

The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m. GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

More information

1.5 Equations of Lines and Planes in 3-D

1.5 Equations of Lines and Planes in 3-D 40 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3-D Recall that given a point P = (a, b, c), one can draw a vector from

More information

8-3 Dot Products and Vector Projections

8-3 Dot Products and Vector Projections 8-3 Dot Products and Vector Projections Find the dot product of u and v Then determine if u and v are orthogonal 1u =, u and v are not orthogonal 2u = 3u =, u and v are not orthogonal 6u = 11i + 7j; v

More information

5 VECTOR GEOMETRY. 5.0 Introduction. Objectives. Activity 1

5 VECTOR GEOMETRY. 5.0 Introduction. Objectives. Activity 1 5 VECTOR GEOMETRY Chapter 5 Vector Geometry Objectives After studying this chapter you should be able to find and use the vector equation of a straight line; be able to find the equation of a plane in

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

Review Sheet for Test 1

Review Sheet for Test 1 Review Sheet for Test 1 Math 261-00 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And

More information

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

Algebra III. Lesson 33. Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids

Algebra III. Lesson 33. Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids Algebra III Lesson 33 Quadrilaterals Properties of Parallelograms Types of Parallelograms Conditions for Parallelograms - Trapezoids Quadrilaterals What is a quadrilateral? Quad means? 4 Lateral means?

More information

Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

More information

Vectors. Objectives. Assessment. Assessment. Equations. Physics terms 5/15/14. State the definition and give examples of vector and scalar variables.

Vectors. Objectives. Assessment. Assessment. Equations. Physics terms 5/15/14. State the definition and give examples of vector and scalar variables. Vectors Objectives State the definition and give examples of vector and scalar variables. Analyze and describe position and movement in two dimensions using graphs and Cartesian coordinates. Organize and

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

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

Math 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t Math 241 Lines and Planes (Solutions) The equations for planes P 1, P 2 and P are P 1 : x 2y + z = 7 P 2 : x 4y + 5z = 6 P : (x 5) 2(y 6) + (z 7) = 0 The equations for lines L 1, L 2, L, L 4 and L 5 are

More information

Math 215 HW #6 Solutions

Math 215 HW #6 Solutions Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T

More information

Why should we learn this? One real-world connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY

Why should we learn this? One real-world connection is to find the rate of change in an airplane s altitude. The Slope of a Line VOCABULARY Wh should we learn this? The Slope of a Line Objectives: To find slope of a line given two points, and to graph a line using the slope and the -intercept. One real-world connection is to find the rate

More information

GCE Mathematics (6360) Further Pure unit 4 (MFP4) Textbook

GCE Mathematics (6360) Further Pure unit 4 (MFP4) Textbook Version 36 klm GCE Mathematics (636) Further Pure unit 4 (MFP4) Textbook The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales 364473 and a

More 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

EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

More information

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

discuss how to describe points, lines and planes in 3 space. Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position

More information

COGNITIVE TUTOR ALGEBRA

COGNITIVE TUTOR ALGEBRA COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,

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

Exam 1 Sample Question SOLUTIONS. y = 2x

Exam 1 Sample Question SOLUTIONS. y = 2x Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can

More information

Skill Builders. (Extra Practice) Volume I

Skill Builders. (Extra Practice) Volume I Skill Builders (Etra Practice) Volume I 1. Factoring Out Monomial Terms. Laws of Eponents 3. Function Notation 4. Properties of Lines 5. Multiplying Binomials 6. Special Triangles 7. Simplifying and Combining

More information

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

Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve

More information

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

... ... . (2,4,5).. ... 12 Three Dimensions ½¾º½ Ì ÓÓÖ Ò Ø ËÝ Ø Ñ So far wehave been investigatingfunctions ofthe form y = f(x), withone independent and one dependent variable Such functions can be represented in two dimensions,

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

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

POINT OF INTERSECTION OF TWO STRAIGHT LINES

POINT OF INTERSECTION OF TWO STRAIGHT LINES POINT OF INTERSECTION OF TWO STRAIGHT LINES THEOREM The point of intersection of the two non parallel lines bc bc ca ca a x + b y + c = 0, a x + b y + c = 0 is,. ab ab ab ab Proof: The lines are not parallel

More information

Geometric description of the cross product of the vectors u and v. The cross product of two vectors is a vector! u x v is perpendicular to u and v

Geometric description of the cross product of the vectors u and v. The cross product of two vectors is a vector! u x v is perpendicular to u and v 12.4 Cross Product Geometric description of the cross product of the vectors u and v The cross product of two vectors is a vector! u x v is perpendicular to u and v The length of u x v is uv u v sin The

More information

Mathematics Placement

Mathematics Placement Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

More information

Orthogonal Projections and Orthonormal Bases

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

More information

www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

More information

Section summaries. d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 1 + y 2. x1 + x 2

Section summaries. d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 1 + y 2. x1 + x 2 Chapter 2 Graphs Section summaries Section 2.1 The Distance and Midpoint Formulas You need to know the distance formula d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 and the midpoint formula ( x1 + x 2, y ) 1 + y 2

More information

Algebra and Geometry Review (61 topics, no due date)

Algebra and Geometry Review (61 topics, no due date) Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

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

LOCUS Definition: The set of all points (and only those points) which satisfy the given geometrical condition(s) (or properties) is called a locus. Eg. The set of points in a plane which are at a constant

More information

Writing the Equation of a Line in Slope-Intercept Form

Writing the Equation of a Line in Slope-Intercept Form Writing the Equation of a Line in Slope-Intercept Form Slope-Intercept Form y = mx + b Example 1: Give the equation of the line in slope-intercept form a. With y-intercept (0, 2) and slope -9 b. Passing

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

Bedford, Fowler: Statics. Chapter 4: System of Forces and Moments, Examples via TK Solver

Bedford, Fowler: Statics. Chapter 4: System of Forces and Moments, Examples via TK Solver System of Forces and Moments Introduction The moment vector of a force vector,, with respect to a point has a magnitude equal to the product of the force magnitude, F, and the perpendicular distance from

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

Solving Simultaneous Equations and Matrices

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

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

Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors 1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number

More information

Dr. Z s Math151 Handout #4.6 [Applied Optimization]

Dr. Z s Math151 Handout #4.6 [Applied Optimization] By Doron Zeilberger Problem Type 4.6.1 : Dr. Z s Math151 Handout #4.6 [Applied Optimization] A farmer wants to fence an area of A square units and then divide it into n + 1 parts by placing n parallel

More information

Use order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS

Use order of operations to simplify. Show all steps in the space provided below each problem. INTEGER OPERATIONS ORDER OF OPERATIONS In the following order: 1) Work inside the grouping smbols such as parenthesis and brackets. ) Evaluate the powers. 3) Do the multiplication and/or division in order from left to right.

More information

Algebra 1 Course Information

Algebra 1 Course Information Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through

More information