8-2 Vectors in the Coordinate Plane

Size: px
Start display at page:

Download "8-2 Vectors in the Coordinate Plane"

Transcription

1 37. ROWING Nadia is rowing across a river at a speed of 5 miles per hour perpendicular to the shore. The river has a current of 3 miles per hour heading downstream. a. At what speed is she traveling? b. At what angle is she traveling with respect to the shore? a. Nadia s rowing can be represented by the vector and the current can be represented by the vector. Add the vectors representing r and c to find the resultant vector, v. The speed at which Nadia is traveling is the magnitude of v. Thus, she is traveling at about 5.8 miles per hour. b. Find θ by constructing a right triangle with the given vectors and using the tangent ratio. Thus, Nadia is traveling at an angle of about 59 with respect to the shore. esolutions Manual - Powered by Cognero Page 1

2 Find the component form of v with the given magnitude and direction angle. 39. v = 4, θ = v = 16, θ = v = 15, θ = 125 esolutions Manual - Powered by Cognero Page 2

3 Find the direction angle of each vector to the nearest tenth of a degree i + 5j Since the vector lies in Quadrant II, θ = ( 68.2 ) or about i 3j Since the vector lies in Quadrant III, θ = or about esolutions Manual - Powered by Cognero Page 3

4 49. So, the direction angle of the vector is Since the vector lies in Quadrant IV, θ = ( 69.4) or about esolutions Manual - Powered by Cognero Page 4

5 53. NAVIGATION An airplane is traveling due east with a speed of 600 miles per hour. The wind blows at 85 miles per hour at an angle of S59 E. a. Determine the speed of the airplane s flight. b. Determine the angle of the airplane s flight. a. Since the airplane is traveling due east with a speed of 600 miles per hour, the component form of the speed v 1 is. Use the magnitude and the direction of the wind v 2 to write this vector in component form. Let θ = 31 since θ is the direction angle that v 2 makes with the positive x-axis. Add the algebraic vectors representing v 1 and v 2 to find the resultant velocity, vector r. Find the magnitude of the resultant. The speed of the airplane s flight is about miles per hour. b. Find the resultant direction angle θ. Since r lies in Quadrant IV, θ = ( 3.7) or Therefore, the angle of the airplane s flight is about S86 E. Determine whether and with the initial and terminal points given are equivalent. If so, prove that =. If not, explain why not. 55. A(3, 5), B(6, 9), C( 4, 4), D( 2, 0) esolutions Manual - Powered by Cognero Page 5

6 Find the magnitude and direction of each vector. For, find the component form. Use the component form of the vector to find the magnitude. Substitute x 2 x 1 = 3 and y 2 y 1 = 4 into the formula for the magnitude of a vector in the coordinate plane. Next, find the direction angle of. For, find the component form. Use the component form of the vector to find the magnitude. Substitute x 2 x 1 = 2 and y 2 y 1 = 4 into the formula for the magnitude of a vector in the coordinate plane. Next, find the direction angle of. esolutions Manual - Powered by Cognero Page 6

7 No; and are not equivalent. The magnitude and direction are not the same for both vectors, so they are not equivalent. 57. A(1, 3), B(0, 10), C(11, 8), D(10, 1) Find the magnitude and direction of each vector. For, find the component form. Use the component form of the vector to find the magnitude. Substitute x 2 x 1 = 1 and y 2 y 1 = 7 into the formula for the magnitude of a vector in the coordinate plane. Next, find the direction angle of. For, find the component form. esolutions Manual - Powered by Cognero Page 7

8 Use the component form of the vector to find the magnitude. Substitute x 2 x 1 = 1 and y 2 y 1 = 7 into the formula for the magnitude of a vector in the coordinate plane. Next, find the direction angle of. Yes; and are equivalent. The magnitude and direction are the same for both vectors, so they are equivalent. 59. NAVIGATION A jet is flying with an air speed of 480 miles per hour at a bearing of N82 E. Because of the wind, the ground speed of the plane is 518 miles per hour at a bearing of N79 E. a. Draw a diagram to represent the situation. b. What are the speed and direction of the wind? c. If the pilot increased the air speed of the plane to 500 miles per hour, what would be the resulting ground speed and direction of the plane? a. Sample answer: Draw a diagram to represent the situation. If the initial bearing of the jet is N82 E but is flying at a bearing of N79 E because of the wind, the angle created by the two vectors is 3. b. The resultant vector r for the ground speed of the jet is the sum of the vector representing the air speed of the jet v 1 and the vector representing the wind v 2, or r = v 1 + v 2. esolutions Manual - Powered by Cognero Page 8

9 Use the air speed and bearing of the jet to write v 1 in component form. Let θ = 8 since the bearing is N82 E. Use the ground speed and bearing of the jet to write r in component form. Let θ = 11 since the bearing is N79 E. Substitute the component forms of r and v 1 into r = v 1 + v 2 and solve for v 2. The component form of the wind is. Find the magnitude of v 2. Find the resultant direction angle θ. The directed angle of the wind is about 44. Thus, the wind is blowing about 46.1 miles per hour at a bearing of about N46 E. esolutions Manual - Powered by Cognero Page 9

10 c. Use the air speed and bearing of the jet to write v 1 in component form. Let θ = 8 since the bearing is N82 E. Substitute the component forms of v 1 and v 2 into r = v 1 + v 2 and solve for r. Find the magnitude of r. Find the resultant direction angle θ. The directed angle of the jet is about 11. Thus, the jet is traveling at a resulting ground speed of about 538 miles per hour at a directed angle of about N79 E. esolutions Manual - Powered by Cognero Page 10

Standardized Test Practice - Cumulative, Chapters What is the value of x in the figure below?

Standardized Test Practice - Cumulative, Chapters What is the value of x in the figure below? 1. What is the value of x in the figure below? 2. A baseball diamond is a square with 90-ft sides. What is the length from 3rd base to 1st base? Round to the nearest tenth. A 22.5 B 23 C 23.5 D 24 Use

More information

Practice Test - Chapter 4

Practice Test - Chapter 4 Find the value of x. Round to the nearest tenth, if necessary. Find the measure of angle θ. Round to the nearest degree, if necessary. 1. An acute angle measure and the length of the hypotenuse are given,

More information

Practice Test - Chapter 4

Practice Test - Chapter 4 Find the value of x. Round to the nearest tenth, if necessary. 1. An acute angle measure and the length of the hypotenuse are given, so the sine function can be used to find the length of the side opposite.

More information

Vectors are used to represent quantities such as force and velocity which have both. and. The magnitude of a vector corresponds to its.

Vectors are used to represent quantities such as force and velocity which have both. and. The magnitude of a vector corresponds to its. Fry Texas A&M University Math 150 Chapter 9 Fall 2014 1 Chapter 9 -- Vectors Remember that is the set of real numbers, often represented by the number line, 2 is the notation for the 2-dimensional plane.

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

Vectors are used to represent quantities such as force and velocity which have both. and. The magnitude of a vector corresponds to its.

Vectors are used to represent quantities such as force and velocity which have both. and. The magnitude of a vector corresponds to its. Fry Texas A&M University Fall 2016 Math 150 Notes Chapter 9 Page 248 Chapter 9 -- Vectors Remember that is the set of real numbers, often represented by the number line, 2 is the notation for the 2-dimensional

More information

4-3 Trigonometric Functions on the Unit Circle

4-3 Trigonometric Functions on the Unit Circle Find the exact value of each trigonometric function, if defined. If not defined, write undefined. 9. sin The terminal side of in standard position lies on the positive y-axis. Choose a point P(0, 1) on

More information

u + v = u - v =, where c Directed Quantities: Quantities such as velocity and acceleration (quantities that involve magnitude as well as direction)

u + v = u - v =, where c Directed Quantities: Quantities such as velocity and acceleration (quantities that involve magnitude as well as direction) Pre-Calculus Section 10.3: Vectors & Their Applications (Part I) 1. Vocabulary (Summary): 4. Algebraic Operations on Vectors: If u = Scalar: A quantity possessing only magnitude (such weight or length

More information

BC VECTOR PROBLEMS. 13. Find the area of the parallelogram having AB and AC as adjacent sides: A(2,1,3), B(1,4,2), C( 3,2,7) 14.

BC VECTOR PROBLEMS. 13. Find the area of the parallelogram having AB and AC as adjacent sides: A(2,1,3), B(1,4,2), C( 3,2,7) 14. For problems 9 use: u (,3) v (3, 4) s (, 7). w =. 3u v = 3. t = 4. 7u = u w (,3,5) 5. wt = t (,, 4) 6. Find the measure of the angle between w and t to the nearest degree. 7. Find the unit vector having

More information

BELLWORK feet

BELLWORK feet BELLWORK 1 A hot air balloon is being held in place by two people holding ropes and standing 35 feet apart. The angle formed between the ground and the rope held by each person is 40. Determine the length

More information

A unit vector in the same direction as a vector a would be a and a unit vector in the

A unit vector in the same direction as a vector a would be a and a unit vector in the In the previous lesson we discussed unit vectors on the positive x-axis (i) and on the positive y- axis (j). What is we wanted to find other unit vectors? There are an infinite number of unit vectors in

More information

Skills Practice Skills Practice for Lesson 14.1

Skills Practice Skills Practice for Lesson 14.1 Skills Practice Skills Practice for Lesson 1.1 Name Date By Air and By Sea Introduction to Vectors Vocabulary Match each term to its corresponding definition. 1. column vector notation a. a quantity that

More information

C) ) cos (cos-1 0.4) 5) A) 0.4 B) 2.7 C) 0.9 D) 3.5 C) - 4 5

C) ) cos (cos-1 0.4) 5) A) 0.4 B) 2.7 C) 0.9 D) 3.5 C) - 4 5 Precalculus B Name Please do NOT write on this packet. Put all work and answers on a separate piece of paper. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the

More information

Chapter 1E - Complex Numbers

Chapter 1E - Complex Numbers Fry Texas A&M University Math 150 Spring 2015 Unit 4 20 Chapter 1E - Complex Numbers 16 exists So far the largest (most inclusive) number set we have discussed and the one we have the most experience with

More information

New concepts: scalars, vectors, unit vectors, vector components, vector equations, scalar product. reading assignment read chap 3

New concepts: scalars, vectors, unit vectors, vector components, vector equations, scalar product. reading assignment read chap 3 New concepts: scalars, vectors, unit vectors, vector components, vector equations, scalar product reading assignment read chap 3 Most physical quantities are described by a single number or variable examples:

More information

4-2 Degrees and Radians

4-2 Degrees and Radians Write each decimal degree measure in DMS form and each DMS measure in decimal degree form to the nearest thousandth. 1. 11.773 First, convert 0. 773 into minutes and seconds. Next, convert 0.38' into seconds.

More information

1.1 Vectors. The length of the vector AB from A(x1,y 1 ) to B(x 2,y 2 ) is

1.1 Vectors. The length of the vector AB from A(x1,y 1 ) to B(x 2,y 2 ) is 1.1 Vectors A vector is a quantity that has both magnitude and direction. Vectors are drawn as directed line segments and are denoted by boldface letters a or by a. The magnitude of a vector a is its length,

More information

10-1 Sequences as Functions. Determine whether each sequence is arithmetic. Write yes or no. 1. 8, 2, 12, 22

10-1 Sequences as Functions. Determine whether each sequence is arithmetic. Write yes or no. 1. 8, 2, 12, 22 Determine whether each sequence is arithmetic. Write yes or no. 1. 8, 2, 12, 22 Subtract each term from the term directly after it. The common difference is 10. 3. 1, 2, 4, 8, 16 Subtract each term from

More information

5-6 Proving Lines Parallel

5-6 Proving Lines Parallel Given the following information, determine which lines, if any, are parallel. State the postulate or theorem that justifies your answer. 1. j k; converse of corresponding angles postulate 3. alternate

More information

Definitions In physics we have two types of measurable quantities: vectors and scalars.

Definitions In physics we have two types of measurable quantities: vectors and scalars. 1 Definitions In physics we have two types of measurable quantities: vectors and scalars. Scalars: have magnitude (magnitude means size) only Examples of scalar quantities include time, mass, volume, area,

More information

5. A triangle has sides represented by the vectors (1, 2) and (5, 6). Determine the vector representing the third side.

5. A triangle has sides represented by the vectors (1, 2) and (5, 6). Determine the vector representing the third side. Vectors EXAM review Problem 1 = 8 and = 1 a) Find the net force, assume that points North, and points East b) Find the equilibrant force 2 = 15, = 7, and the angle between and is 60 What is the magnitude

More information

3-1 Constant Rate of Change

3-1 Constant Rate of Change Determine whether the relationship between the two quantities shown in the table or graph is linear. If so, find the constant rate of change. If not, explain your reasoning. 1. Analyze the table. The rate

More information

6-2 Matrix Multiplication, Inverses and Determinants

6-2 Matrix Multiplication, Inverses and Determinants Find AB and BA, if possible. 1. A = A = ; A is a 1 2 matrix and B is a 2 2 matrix. Because the number of columns of A is equal to the number of rows of B, AB exists. To find the first entry of AB, find

More information

Newton 3 & Vectors. Action/Reaction. You Can OnlyTouch as Hard as You Are Touched 9/7/2009

Newton 3 & Vectors. Action/Reaction. You Can OnlyTouch as Hard as You Are Touched 9/7/2009 Newton 3 & Vectors Action/Reaction When you lean against a wall, you exert a force on the wall. The wall simultaneously exerts an equal and opposite force on you. You Can OnlyTouch as Hard as You Are Touched

More information

Congruence Axioms. Data Required for Solving Oblique Triangles

Congruence Axioms. Data Required for Solving Oblique Triangles Math 335 Trigonometry Sec 7.1: Oblique Triangles and the Law of Sines In section 2.4, we solved right triangles. We now extend the concept to all triangles. Congruence Axioms Side-Angle-Side SAS Angle-Side-Angle

More information

Relative Velocities In Two Dimensions

Relative Velocities In Two Dimensions Relative Velocities In Two Dimensions The heading is the angle of the moving body the direction the object is pointing. The resultant velocity is typically the velocity of the object relative to the ground.

More information

OpenStax-CNX module: m Vectors. OpenStax College. Abstract

OpenStax-CNX module: m Vectors. OpenStax College. Abstract OpenStax-CNX module: m49412 1 Vectors OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section you will: Abstract View vectors

More information

Vectors. An Introduction

Vectors. An Introduction Vectors An Introduction There are two kinds of quantities Scalars are quantities that have magnitude only, such as position speed time mass Vectors are quantities that have both magnitude and direction,

More information

8-6 Solving Rational Equations and Inequalities. Solve each equation. Check your solution. ANSWER: 11 ANSWER: 9 ANSWER: 7 ANSWER: 3 ANSWER: 8

8-6 Solving Rational Equations and Inequalities. Solve each equation. Check your solution. ANSWER: 11 ANSWER: 9 ANSWER: 7 ANSWER: 3 ANSWER: 8 Solve each equation. Check your solution. 1. 11 2. 9 3. 7 4. 3 5. 8 6. 5 esolutions Manual - Powered by Cognero Page 1 7. 14 8. 14 9. CCSS STRUCTURE Sara has 10 pounds of dried fruit selling for $6.25

More information

Mid-Chapter Quiz: Lessons 10-1 through Refer to. 1. Name the circle. SOLUTION: The center of the circle is A. Therefore, the circle is ANSWER:

Mid-Chapter Quiz: Lessons 10-1 through Refer to. 1. Name the circle. SOLUTION: The center of the circle is A. Therefore, the circle is ANSWER: Refer to. 1. Name the circle. The center of the circle is A. Therefore, the circle is 2. Name a diameter. ; since is a chord that passes through the center, it is a diameter. 3. Name a chord that is not

More information

4-6 Inverse Trigonometric Functions

4-6 Inverse Trigonometric Functions Find the exact value of each expression, if it exists 1 sin 1 0 with a y-coordinate of 0 3 arcsin When t = 0, sin t = 0 Therefore, sin 1 0 = 0 2 arcsin When t =, sin t = Therefore, arcsin = 4 sin 1 When

More information

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. 3. u = 9, 3, v = 1, 3 Since, u and v are orthogonal. 6. u = 11i + 7j; v = 7i + 11j Write u and v in component form as Since, u

More information

Physics 12. Chapter 1: Vector Analysis in Two Dimensions

Physics 12. Chapter 1: Vector Analysis in Two Dimensions Physics 12 Chapter 1: Vector Analysis in Two Dimensions 1. Definitions When studying mechanics in Physics 11, we have realized that there are two major types of quantities that we can measure for the systems

More information

12-4 Law of Sines. Find the area of ABC to the nearest tenth, if necessary. SOLUTION: Substitute c = 7, b = 8 and A = 86º in the area. formula.

12-4 Law of Sines. Find the area of ABC to the nearest tenth, if necessary. SOLUTION: Substitute c = 7, b = 8 and A = 86º in the area. formula. Find the area of ABC to the nearest tenth, if necessary. 3. A = 40, b = 11 cm, c = 6 cm Substitute c = 6, b = 11 and A = 40º in the area 1. Substitute c = 7, b = 8 and A = 86º in the area 4. B = 103, a

More information

DATE: MATH ANALYSIS 2 CHAPTER 12: VECTORS & DETERMINANTS

DATE: MATH ANALYSIS 2 CHAPTER 12: VECTORS & DETERMINANTS NAME: PERIOD: DATE: MATH ANALYSIS 2 MR. MELLINA CHAPTER 12: VECTORS & DETERMINANTS Sections: v 12.1 Geometric Representation of Vectors v 12.2 Algebraic Representation of Vectors v 12.3 Vector and Parametric

More information

Trigonometry Test 3 Practice Chapters 5 and 6 NON-CALCULATOR PORTION

Trigonometry Test 3 Practice Chapters 5 and 6 NON-CALCULATOR PORTION NON-CALCULATOR PORTION Find four solutions to each of the following; write your answer in 1. 2. 3. 4. 5. 6. radians: Find the value of each of the following: 7. ( ) 8. 9. ( ) 10. 11. 12. 13. ( ) Find four

More information

0-8 Area. Find the area of each figure. 1. SOLUTION: The area of the rectangle is 6 square centimeters. 2. SOLUTION:

0-8 Area. Find the area of each figure. 1. SOLUTION: The area of the rectangle is 6 square centimeters. 2. SOLUTION: Find the area of each figure. 1. The area of the rectangle is 6 square centimeters. 2. The area of the square is 36 square inches. 3. The area of the parallelogram is 120 square meters. esolutions Manual

More information

b g 6. P 2 4 π b g b g of the way from A to B. LATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME OF THE CHAPTER TEST ON ASSIGNMENT DUE

b g 6. P 2 4 π b g b g of the way from A to B. LATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME OF THE CHAPTER TEST ON ASSIGNMENT DUE A Trig/Math Anal Name No LATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME OF THE CHAPTER TEST ON HW NO. SECTIONS (Brown Book) ASSIGNMENT DUE V 1 1 1/1 Practice Set A V 1 3 Practice Set B #1 1 V B 1

More information

Find the length of an arc that subtends a central angle of 45 in a circle of radius 8 m. Round your answer to 3 decimal places.

Find the length of an arc that subtends a central angle of 45 in a circle of radius 8 m. Round your answer to 3 decimal places. Chapter 6 Practice Test Find the radian measure of the angle with the given degree measure. (Round your answer to three decimal places.) 80 Find the degree measure of the angle with the given radian measure:

More information

Math Review -- Conceptual Solutions

Math Review -- Conceptual Solutions Math Review Math Review -- Conceptual Solutions 1.) Is three plus four always equal to seven? Explain. Solution: If the numbers are scalars written in base 10, the answer is yes (if the numbers are in

More information

Study Guide and Review. 11. Find EG if G is the incenter of.

Study Guide and Review. 11. Find EG if G is the incenter of. 11. Find EG if G is the incenter of. By the Incenter Theorem, since G is equidistant from the sides of Pythagorean Theorem., EG = FG. Find FG using the Since length cannot be negative, use only the positive

More information

United Arab Emirates University

United Arab Emirates University United Arab Emirates University University Foundation Program - Math Program ALGEBRA - COLLEGE ALGEBRA - TRIGONOMETRY Practice Questions 1. What is 2x 1 if 4x + 8 = 6 + x? A. 2 B. C. D. 4 E. 2. What is

More information

2-6 Analyzing Functions with Successive Differences

2-6 Analyzing Functions with Successive Differences Graph each set of ordered pairs. Determine whether the ordered pairs represent a linear function, a quadratic function, or an exponential function. 1. ( 2, 8), ( 1, 5), (0, 2), (1, 1) linear 3. ( 3, 8),

More information

What is Relative Motion

What is Relative Motion RELATIVE MOTION What is Relative Motion Strictly speaking all motion is relative to something. Usually that something is a reference point that is assumed to be at rest (i.e. the earth). Motion can be

More information

Polar Coordinates; Vectors

Polar Coordinates; Vectors Chapter 10 Polar Coordinates; Vectors 10.R Chapter Review 1. 3, 6 x = 3cos 6 = 3 3. 4, 3 x = 4cos 3 = y = 3sin 6 = 3 3 3, 3 y =4sin 3 = 3 (, 3) 3., 4 3 x = cos 4 3 =1 4. 1, 5 4 x = 1cos 5 4 = y = sin 4

More information

Spring 2010 Physics 141 Practice Exam II Phy141_mt1b.pdf

Spring 2010 Physics 141 Practice Exam II Phy141_mt1b.pdf 1. (15 points) You are given two vectors: A has length 10. and an angle of 60. o (with respect to the +x axis). B has length 10. and an angle of 200. o (with respect to the +x axis). a) Calculate the components

More information

Physics 12 Unit 1: Kinematics Notes. Name: What you will be able to do by the end of this unit:

Physics 12 Unit 1: Kinematics Notes. Name: What you will be able to do by the end of this unit: Physics 12 Unit 1: Kinematics Notes. Name: What you will be able to do by the end of this unit: B1. Perform vector analysis in one or two dimensions identify scalars and vectors resolve a vector into two

More information

Homework due Nov 28 Physics

Homework due Nov 28 Physics Homework due Nov 28 Physics Name Base your answers to questions 1 through 4 on the information and vector diagram below and on your knowledge of physics. A hiker starts at point P and walks 2.0 kilometers

More information

SB Ch 6 May 15, 2014

SB Ch 6 May 15, 2014 Warm Up 1 Chapter 6: Applications of Trig: Vectors Section 6.1 Vectors in a Plane Vector: directed line segment Magnitude is the length of the vector Direction is the angle in which the vector is pointing

More information

Chapter 7 Test. 2. In triangle ABC, A = 60, and side c = 20 ft. How many triangles can be formed if side a = 16 ft? A) 0 B) 1 C) 2 D) 3

Chapter 7 Test. 2. In triangle ABC, A = 60, and side c = 20 ft. How many triangles can be formed if side a = 16 ft? A) 0 B) 1 C) 2 D) 3 Name Chapter 7 Test 1. Solve the triangle using the law of sines. Round to the nearest tenth. side a = 12 m A = 19 B = 79 What are the lengths of sides b and c? A) b = 35.5 m, c = 35.9 m C) b = 36.2 m,

More information

Appendix D: Algebra and Trig Review

Appendix D: Algebra and Trig Review Appendix D: Algebra and Trig Review Find the domains of the following functions. x+2 x 2 5x+4 3 x 4 + x 2 9 7 x If f(x) = x 3, find f(8+h) f(8) h and simplify by rationalizing the numerator. 1 Converting

More information

Kinematics. Vector solutions. Vectors

Kinematics. Vector solutions. Vectors Kinematics Study of motion Accelerated vs unaccelerated motion Translational vs Rotational motion Vector solutions required for problems of 2- directional motion Vector solutions Possible solution sets

More information

Math 370 Exam 3 Review Name

Math 370 Exam 3 Review Name Math 70 Exam Review Name The following problems will give you an idea of the concepts covered on the exam. Note that the review questions may not be formatted like those on the exam. You should complete

More information

Student Content Brief Advanced Level

Student Content Brief Advanced Level Student Content Brief Advanced Level Vectors Background Information Physics and Engineering deal with quantities that have both size and direction. These physical quantities have a special math language

More information

a) Plot and label the 3 points. [3] b) Rewrite point A using a negative angle and negative radius.

a) Plot and label the 3 points. [3] b) Rewrite point A using a negative angle and negative radius. Analysis Summer HW problem Set: Compiled 207 Students bridging up to Analysis from Alg2/TrigA will need to learn a few topics on their own this summer. These are topics that other Analysis students learned

More information

Chapter 3. Vectors. θ that the vector forms with i ˆ is 15. I. Vectors and Scalars

Chapter 3. Vectors. θ that the vector forms with i ˆ is 15. I. Vectors and Scalars Chapter 3. Vectors I. Vectors and Scalars 1. What type of quantity does the odometer of a car measure? a) vector; b) scalar; c) neither scalar nor vector; d) both scalar and vector. 2. What type of quantity

More information

Student Exploration: Vectors

Student Exploration: Vectors Name: Date: Student Exploration: Vectors Vocabulary: component, dot product, magnitude, resultant, scalar, unit vector notation, vector Prior Knowledge Question (Do this BEFORE using the Gizmo.) An airplane

More information

Chapter 1: Trigonometric Functions 1. Find (a) the complement and (b) the supplement of 61. Show all work and / or support your answer.

Chapter 1: Trigonometric Functions 1. Find (a) the complement and (b) the supplement of 61. Show all work and / or support your answer. Trig Exam Review F07 O Brien Trigonometry Exam Review: Chapters,, To adequately prepare for the exam, try to work these review problems using only the trigonometry knowledge which you have internalized

More information

2-6 Algebraic Proof. State the property that justifies each statement. 1. If m 1 = m 2 and m 2 = m 3, then m 1 = m 3. SOLUTION:

2-6 Algebraic Proof. State the property that justifies each statement. 1. If m 1 = m 2 and m 2 = m 3, then m 1 = m 3. SOLUTION: State the property that justifies each 1. If m 1 = m 2 and m 2 = m 3, then m 1 = m 3. There are two parts to the hypotheses. "If m 1 = m 2 and m 2 = m 3, then m 1 = m 3. "The end of the first part of the

More information

Study Guide and Review - Chapter 5. Solve each inequality. Then graph it on a number line. 11. w 4 > 9 SOLUTION: The solution set is {w w > 13}.

Study Guide and Review - Chapter 5. Solve each inequality. Then graph it on a number line. 11. w 4 > 9 SOLUTION: The solution set is {w w > 13}. Solve each inequality. Then graph it on a number line. 11. w 4 > 9 The solution set is {w w > 13}. 13. 6 + h < 1 The solution set is {h h < 5}. 15. 13 p 15 The solution set is {p p 2}. 17. FIELD TRIP A

More information

Find graphically, using scaled diagram, following vectors (both magnitude and direction):

Find graphically, using scaled diagram, following vectors (both magnitude and direction): 1 HOMEWORK 1 on VECTORS: use ruler and protractor, please!!! 1. v 1 = 3m/s, E and v = 4m/s, 3 Find graphically, using scaled diagram, following vectors (both magnitude and direction): a. v = v 1 + v b.

More information

Learning to Fly. Denise Russo. September 17, 2010

Learning to Fly. Denise Russo. September 17, 2010 Learning to Fly Denise Russo September 17, 2010 Content Area: Trigonometry Grade Level: 11-12 Date: Sept. 17, 2010 Text Selection: Trigonometry Section 7.5: Vectors Author(s): Margaret L. Lial, Charles

More information

3-1 Solving Systems of Equations. Solve each system of equations by using a table. 1. ANSWER: (3, 5) ANSWER: (2, 7)

3-1 Solving Systems of Equations. Solve each system of equations by using a table. 1. ANSWER: (3, 5) ANSWER: (2, 7) Solve each system of equations by using a table. 1. 9. CCSS MODELING Refer to the table below. (3, 5) 2. (2, 7) Solve each system of equations by graphing. 3. a. Write equations that represent the cost

More information

Scalar Quantities - express only magnitude ie. time, distance, speed

Scalar Quantities - express only magnitude ie. time, distance, speed Chapter 6 - Vectors Scalar Quantities - express only magnitude ie. time, distance, speed Vector Quantities - express magnitude and direction. ie. velocity 80 km/h, 58 displacement 10 km (E) acceleration

More information

5-3 Solving Multi-Step Inequalities. Solve each inequality. Graph the solution on a number line b 1 11 SOLUTION: The solution set is {b b 2}.

5-3 Solving Multi-Step Inequalities. Solve each inequality. Graph the solution on a number line b 1 11 SOLUTION: The solution set is {b b 2}. Solve each inequality. Graph the solution on a number line. 12. 5b 1 11 14. 9 m + 7 The solution set is {b b 2}. {b b 2} The solution set is {m m 40}. 13. 21 > 15 + 2a {m m 40} 15. 13 > 6 The solution

More information

Unit 5 RATIONAL FUNCTIONS. A function with a variable in the denominator Parent function 1 x Graph is a hyperbola

Unit 5 RATIONAL FUNCTIONS. A function with a variable in the denominator Parent function 1 x Graph is a hyperbola Unit 5 RATIONAL FUNCTIONS A function with a variable in the denominator Parent function 1 x Graph is a hyperbola A direct variation is a relationship between two variables x and y that can be written in

More information

Pre-Calculus 40 Final Outline/Review:

Pre-Calculus 40 Final Outline/Review: 2016-2017 Pre-Calculus 40 Final Outline/Review: Non-Calculator Section: 16 multiple choice (32 pts) and 6 open ended (24 pts). Calculator Section: 8 multiple choice (16 pts) and 11 open ended (36 pts).

More information

College Prep Math Final Exam Review Packet

College Prep Math Final Exam Review Packet College Prep Math Final Exam Review Packet Name: Date of Exam: In Class 1 Directions: Complete each assignment using the due dates given by the calendar below. If you are absent from school, you are still

More information

8-1 Geometric Mean. SOLUTION: We have the diagram as shown.

8-1 Geometric Mean. SOLUTION: We have the diagram as shown. 25. CCSS MODELING Makayla is using a book to sight the top of a waterfall. Her eye level is 5 feet from the ground and she is a horizontal distance of 28 feet from the waterfall. Find the height of the

More information

10-3 Arcs and Chords. ALGEBRA Find the value of x.

10-3 Arcs and Chords. ALGEBRA Find the value of x. ALGEBRA Find the value of x. 1. Arc ST is a minor arc, so m(arc ST) is equal to the measure of its related central angle or 93. and are congruent chords, so the corresponding arcs RS and ST are congruent.

More information

VECTORS. Section 6.3 Precalculus PreAP/Dual, Revised /11/ :41 PM 6.3: Vectors in the Plane 1

VECTORS. Section 6.3 Precalculus PreAP/Dual, Revised /11/ :41 PM 6.3: Vectors in the Plane 1 VECTORS Section 6.3 Precalculus PreAP/Dual, Revised 2017 Viet.dang@humbleisd.net 10/11/2018 11:41 PM 6.3: Vectors in the Plane 1 DEFINITIONS A. Vector is used to indicate a quantity that has both magnitude

More information

5-5 The Triangle Inequality

5-5 The Triangle Inequality Is it possible to form a triangle with the given side lengths? If not, explain why not. 1. 5 cm, 7 cm, 10 cm Yes; 5 + 7 > 10, 5 + 10 > 7, and 7 + 10 > 5 3. 6 m, 14 m, 10 m Yes; 6 + 14 > 10, 6 + 10 > 14,

More information

10-1 Sequences as Functions. Determine whether each sequence is arithmetic. Write yes or no , 3, 0, 3, 9

10-1 Sequences as Functions. Determine whether each sequence is arithmetic. Write yes or no , 3, 0, 3, 9 Determine whether each sequence is arithmetic. Write yes or no. 22. 9, 3, 0, 3, 9 Find the next four terms of each arithmetic sequence. Then graph the sequence. 26. 10, 2, 6, 14, There is no common difference.

More information

Test # 3 Review Math Name (6.5 to 6.7, 10.1 to 10.3,and 10.5)

Test # 3 Review Math Name (6.5 to 6.7, 10.1 to 10.3,and 10.5) Test # Review Math 14 Name (6.5 to 6.7, 10.1 to 10.,and 10.5) Date: MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the product of the complex

More information

Polar Coordinates; Vectors

Polar Coordinates; Vectors 10.5 The Dot Product 1. v i, w i+ (a) v w 1(1) + ( 1)(1) 1 1 0 (b) cos v w 0 1 + ( 1) 1 + 1 0 0 0 90º (c) The vectors are orthogonal.. v i +, w i+ (a) v w 1( 1) +1(1) 1 + 1 0 (b) cos v w 0 1 +1 ( 1) +

More information

Mid-Chapter Quiz: Lessons 1-1 through 1-4

Mid-Chapter Quiz: Lessons 1-1 through 1-4 Determine whether each relation represents y as a function of x. 1. 3x + 7y = 21 This equation represents y as a function of x, because for every x-value there is exactly one corresponding y-value. function

More information

5-4 Sum and Difference Identities

5-4 Sum and Difference Identities Find the exact value of each trigonometric expression. 1. cos 75 Write 75 as the sum or difference of angle measures with cosines that you know. 3. sin Write as the sum or difference of angle measures

More information

Mt. Douglas Secondary

Mt. Douglas Secondary Foundations of Math 11 Section 3.4 pplied Problems 151 3.4 pplied Problems The Law of Sines and the Law of Cosines are particularly useful for solving applied problems. Please remember when using the Law

More information

Unit 5 RATIONAL FUNCTIONS. A function with a variable in the denominator Parent function 1 x Graph is a hyperbola

Unit 5 RATIONAL FUNCTIONS. A function with a variable in the denominator Parent function 1 x Graph is a hyperbola Unit 5 RATIONAL FUNCTIONS A function with a variable in the denominator Parent function 1 x Graph is a hyperbola I will be following the Alg 2 book in this Unit Ch 5 Sections 1-5 Use the Practice Packet

More information

G.SRT.D.11: Vectors. Regents Exam Questions G.SRT.D.11: Vectors

G.SRT.D.11: Vectors. Regents Exam Questions G.SRT.D.11: Vectors Regents Exam Questions G.SRT.D.11: Vectors www.jmap.org Name: G.SRT.D.11: Vectors 1 The accompanying diagram shows a resultant force vector, R. 4 Two equal forces act on a body at an angle of 80. If the

More information

Math 370 Exam 3 Review Name

Math 370 Exam 3 Review Name Math 370 Exam 3 Review Name The following problems will give you an idea of the concepts covered on the exam. Note that the review questions may not be formatted like those on the exam. You should complete

More information

Precalculus A - Final Exam Review Fall, 2014

Precalculus A - Final Exam Review Fall, 2014 Name: Precalculus A - Final Exam Review Fall, 2014 Period: Find the measures of two angles, one positive and one negative, that are coterminal with the given angle. 1) 85 2) -166 3) 3 Convert the radian

More information

2-4 Zeros of Polynomial Functions

2-4 Zeros of Polynomial Functions Write a polynomial function of least degree with real coefficients in standard form that has the given zeros. 33. 2, 4, 3, 5 Using the Linear Factorization Theorem and the zeros 2, 4, 3, and 5, write f

More information

Find a vector equation for the line through R parallel to the line (PQ) (Total 6 marks)

Find a vector equation for the line through R parallel to the line (PQ) (Total 6 marks) 1. The points P( 2, 4), Q (3, 1) and R (1, 6) are shown in the diagram below. (a) Find the vector PQ. (b) Find a vector equation for the line through R parallel to the line (PQ). 2. The position vector

More information

Two-Dimensional Kinematics: Heading North (Solutions)

Two-Dimensional Kinematics: Heading North (Solutions) Two-Dimensional Kinematics: Heading North (Solutions) You are the navigator of a TWA flight scheduled to fly from New Orleans due north to St. Louis, a distance of 673 miles. Your instruments tell you

More information

Vector Supplement Part 1: Vectors

Vector Supplement Part 1: Vectors Vector Supplement Part 1: Vectors A vector is a quantity that has both magnitude and direction. Vectors are drawn as directed line segments and are denoted by boldface letters a or by a. The magnitude

More information

Name: Period Score /27 Version: A

Name: Period Score /27 Version: A Name: Period Score /27 Version: A Math 11 - Adult Education - Trigonometry Short Answer Show any work and the answer in the space provided. 1. (1 point) What is the reference angle for 215 in standard

More information

3) sin 265 cos 25 - cos 265 sin 25 C) Find the exact value by using a sum or difference identity. 4) sin 165 C) - 627

3) sin 265 cos 25 - cos 265 sin 25 C) Find the exact value by using a sum or difference identity. 4) sin 165 C) - 627 Bonus Assignment Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the given information to find the exact value of the expression. 1) sin

More information

9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b

9.1 VECTORS. A Geometric View of Vectors LEARNING OBJECTIVES. = a, b vectors and POLAR COORDINATES LEARNING OBJECTIVES In this section, ou will: View vectors geometricall. Find magnitude and direction. Perform vector addition and scalar multiplication. Find the component

More information

Inquiry Lab: Unit Rates

Inquiry Lab: Unit Rates Work with a partner to solve. 1. Travis drove 129 miles in 3 hours. He drove at a constant speed. How many miles did he drive in 1 hour? Step 1 The bar diagram represents 129 miles. Divide the bar diagram

More information

Vectors and Kinematics Notes 1 Review

Vectors and Kinematics Notes 1 Review Velocity is defined as the change in displacement with respect to time. Vectors and Kinematics Notes 1 Review Note that this formula is only valid for finding constant velocity or average velocity. Also,

More information

10-2 Simplifying Radical Expressions. Simplify each expression. SOLUTION: 4. SOLUTION: SOLUTION: SOLUTION: 10. MULTIPLE CHOICE Which expression is

10-2 Simplifying Radical Expressions. Simplify each expression. SOLUTION: 4. SOLUTION: SOLUTION: SOLUTION: 10. MULTIPLE CHOICE Which expression is 2. Simplify each expression. 10. MULTIPLE CHOICE Which expression is equivalent to? A B 4. C D 6. 8. 12. The correct choice is D. Simplify each expression. esolutions Manual - Powered by Cognero Page 1

More information

CHAPTER 2: VECTOR COMPONENTS DESCRIBE MOTION IN TWO DIMENSIONS

CHAPTER 2: VECTOR COMPONENTS DESCRIBE MOTION IN TWO DIMENSIONS CHAPTER 2: VECTOR COMPOETS DESCRIBE MOTIO I TWO DIMESIOS 2.1 Vector Methods in One Dimension Vectors may be pictured with sketches in which arrows represent quantities such as displacement, force and velocity.

More information

4-7 Inverse Linear Functions

4-7 Inverse Linear Functions Find the inverse of each relation. 1. {(4, 15), ( 8, 18), ( 2, 16.5), (3, 15.25)} {( 15, 4), ( 18, 8), ( 16.5, 2), ( 15.25, 3)} 2. {(11.8, 3), (3.7, 0), (1, 1), ( 12.5, 6)} Graph the inverse of each relation.

More information

3-4 Systems of Equations in Three Variables. Solve each system of equations. SOLUTION:

3-4 Systems of Equations in Three Variables. Solve each system of equations. SOLUTION: Solve each system of equations. 3. Multiply the second equation by 2 and add with the third equation. Multiply the first equation by 2 and add with the second equation. Solve the fifth and fourth equations.

More information

1) SSS 2) SAS 3) ASA 4) AAS Never: SSA and AAA Triangles with no right angles.

1) SSS 2) SAS 3) ASA 4) AAS Never: SSA and AAA Triangles with no right angles. NOTES 6 & 7: TRIGONOMETRIC FUNCTIONS OF ANGLES AND OF REAL NUMBERS Name: Date: Mrs. Nguyen s Initial: LESSON 6.4 THE LAW OF SINES Review: Shortcuts to prove triangles congruent Definition of Oblique Triangles

More information

SOLUTION: The domain of a square root function only includes values for which the radicand is nonnegative.

SOLUTION: The domain of a square root function only includes values for which the radicand is nonnegative. 19. Graph each function. State the domain and range. 21. The domain of a square root function only includes values for which the radicand is nonnegative. esolutions Manual - Powered by Cognero Page 1 23.

More information

Final Exam Review / AP Calculus AB

Final Exam Review / AP Calculus AB Chapter : Final Eam Review / AP Calculus AB Use the graph to find each limit. 1) lim f(), lim f(), and lim π - π + π f 5 4 1 y - - -1 - - -4-5 ) lim f(), - lim f(), and + lim f 8 6 4 y -4 - - -1-1 4 5-4

More information

CHAPTER 3 KINEMATICS IN TWO DIMENSIONS; VECTORS

CHAPTER 3 KINEMATICS IN TWO DIMENSIONS; VECTORS CHAPTER 3 KINEMATICS IN TWO DIMENSIONS; VECTORS OBJECTIVES After studying the material of this chapter, the student should be able to: represent the magnitude and direction of a vector using a protractor

More information

Unit two review (trig)

Unit two review (trig) Class: Date: Unit two review (trig) Multiple Choice Identify the choice that best completes the statement or answers the question. 1. What is the reference angle for 15 in standard position? A 255 C 345

More information