Skills Practice Skills Practice for Lesson 14.1

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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 has magnitude, or length. components b. a quantity that has both magnitude 3. equivalent vectors and direction. initial point of a vector c. the point at which a vector begins 5. magnitude of a vector d. the point at which a vector ends. negative vector e. a vector with the initial point located at the origin 7. position vector. scalar 9. terminal point of a vector 10. vector 11. vector notation f. a method of epressing a vector using a capital letter and an arrow that includes the vector's components g. the horizontal and vertical influence on a vector h. the length of a vector i. vectors with the same magnitude and direction 010 Carnegie Learning, Inc. j. a vector with the same magnitude as a given vector, but traveling in the opposite direction k. a vector written in matri fashion 1 Chapter 1 l Skills Practice 1009

2 Problem Set Use vector notation to represent each vector. 1. X. (0, 0) (0, 5) P (7, ) (0, 0) X 0, 5 3. (0, 0) M. (0, 0) K (, ) (, 3) 5. (0, 0). ( 5, 5) Z (0, 5) J (0, 0) 1 Determine the magnitude of 7. A (, ) AB _ AB AB. B (, 5). A (, 5) B (, 1) 010 Carnegie Learning, Inc Chapter 1 Skills Practice

3 Name Date 9. A (, 5) 10. B (7, ) B (, 0) A (15, 5) 11. B (, 7) 1. A (1, 1) B (, ) A (, 5) Use v on each grid shown to draw three equivalent vectors. Locate one of the vectors such that its initial point is at the origin. 010 Carnegie Learning, Inc y 1. v v y 0 1 Chapter 1 Skills Practice 1011

4 15. y 1. y 0 v v y v 1. y v Draw w equal to v on each grid shown and locate the initial point at the origin. Then draw w with the initial point at the origin. 19. w y 0 w v 0. v y Carnegie Learning, Inc. 101 Chapter 1 Skills Practice

5 Name Date 1. y. y v 0 0 v 3. y. y 0 v 0 v Use column vector notation to represent each vector. 010 Carnegie Learning, Inc. 5. y. 0 Q R y T 0 1 P QR ( 1 ) Chapter 1 Skills Practice 1013

6 7. y. y V 0 L Z 0 S 9. y 30. y C A K 0 J 0 1 Compare the vectors without graphing. 31. YU 7, and FL, 7 YU 7, : The translation of Y to U is 7 to the right and down. _ FL, 7 : The translation of F to L is to the left and 7 up. 010 Carnegie Learning, Inc. 3. _ PS ( 1 ) and _ V T, Chapter 1 Skills Practice

7 Name 33. RM, and IP ( ) Date 3. KS 1, 5 and JV 1, _ TV ( 0 5 ) and NB ( 0 5 ) 3. JK 5, and XQ ( 5 ) Use the given information to answer each question. 37. An airplane is traveling due east. The aircraft encounters a headwind blowing due south. The vector XZ represents the velocity of the airplane traveling due east. _ The vector ZQ represents the wind blowing due south. Draw vector XQ with initial _ point X and terminal point Q. Then compare vectors XZ and XQ with respect to speed. 010 Carnegie Learning, Inc. X Z Q Vector XQ is slightly faster than vector XZ because it is the hypotenuse of a right triangle. Because side XQ is longer than side XZ, it would follow that _ vector XQ is faster than vector XZ. 1 Chapter 1 Skills Practice 1015

8 3. A sailboat is traveling due south. The boat encounters a current heading due east. The vector SP represents the velocity of the sailboat traveling due south. The vector PM represents the wind blowing due east. Draw vector SM with initial point S and terminal point M. Compare vectors SP and SM with respect to direction. S P M 39. A rowboat is moving 0 feet per minute downstream due south and encounters a current. The current is moving 7 feet per minute due north. Describe the effect the current has on the boat with respect to speed and direction. Use vectors to illustrate this situation A boat is moving kilometers per hour due north and encounters a current. The current is moving 1 kilometers per hour due south. Describe the effect the current has on the boat with respect to speed and direction. Use vectors to illustrate this situation. 010 Carnegie Learning, Inc. 101 Chapter 1 Skills Practice

9 Name Date 1. An airplane traveling due east encounters a tailwind. Use vectors to represent the airplane velocity, the tailwind velocity, and the vector resulting from the airplane encountering a tailwind.. An airplane traveling due south encounters a crosswind that is blowing due west. Use vectors to represent the airplane velocity, the crosswind velocity, and the vector resulting from the airplane encountering a crosswind. 010 Carnegie Learning, Inc. 1 Chapter 1 Skills Practice 1017

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11 Skills Practice Skills Practice for Lesson 1. Name Date Adding Vectors Adding and Subtracting Vectors Vocabulary Write the term that best completes each sentence. 1. A(n) is a single vector that is formed by combining two or more vectors.. The operation of adding two or more vectors is called. 3. A vector resulting from vector addition is called a(n).. The states that the vector sum X Y is the vector drawn from the initial point of vector X to the terminal point of vector Y if the terminal point of vector X and the initial point of vector Y are the same point. 5. The states that the vector sum H J is the vector drawn as the diagonal of the parallelogram formed when the initial points of the vectors are the same point and the terminal points are used to create the missing two sides of the parallelogram.. A vector of length zero that has all components with a value of 0 is called a(n). 010 Carnegie Learning, Inc. 1 Chapter 1 Skills Practice 1019

12 Problem Set Graph vectors A and B and use the Triangle Rule for Vector Addition to determine the resultant vector, C, where C A B. Add the components to check your answer. 1. A 1,, B, 3. A,, B, 1 A C B C 1, 5 Check: 1,, 3 1, 5 3. A 0,, B 1, 1. A,, B, Carnegie Learning, Inc. 100 Chapter 1 Skills Practice

13 Name Date 5. A ( 3 3 ), B ( 1 1 ). A ( 1 ), B ( ) Graph vectors A and B and use the Parallelogram Rule for Vector Addition to determine the resultant vector, C, where C A B. Add the components to check your answer. 7. A 3, 0, B, 3. A 1, 1, B 0, 010 Carnegie Learning, Inc. C A B 1 C 5, 3 Check: 3, 0, 3 5, 3 Chapter 1 Skills Practice 101

14 9. A 3,, B, A 5, 3, B, 11. A ( 1 ), B ( 3 ) 1. A ( 3 1 ), B ( 0 ) Carnegie Learning, Inc. 10 Chapter 1 Skills Practice

15 Name Date Graph vectors A and B and use the Triangle Rule for Vector Addition to determine the resultant vector, C, where C B A. Subtract the components to check your answer. 13. A 3,, B 1, 1. A 0,, B 3, C B A 15. C, Check: 1, 3,, A 1,, B 3, 1 1. A,, B 1, 010 Carnegie Learning, Inc. 1 Chapter 1 Skills Practice 103

16 17. A ( 3 ), B ( 1 ) 1. A ( 1 ), B ( 0 ) Graph vectors A and B and use the Parallelogram Rule for Vector Addition to determine the resultant vector, C, where C B A. Subtract the components to check your answer. 19. A, 1, B, 3 0. A 0,, B 1, 1 1 B A C 010 Carnegie Learning, Inc., 3, 1, C, 10 Chapter 1 Skills Practice

17 Name Date 1. A 1, 1, B 3,. A 3,, B 0, 3. A ( 1 ), B ( 3 ). A ( 3 1 ), B ( 0 ) 010 Carnegie Learning, Inc. 1 Chapter 1 Skills Practice 105

18 Use the properties of vectors to show that each statement is true. 5. Given F 1, 3 and G, 0, use algebra to show F G G F G 1, 3, 0 5, 3 G F, 0 1, 3 5, 3 F.. Given K,, L 1,, and M, 1, use algebra to show ( K L ) M K ( L M ). 7. Given Y,, use algebra to show Y ( Y ) Given E, 3, use algebra to show E 0 E. 010 Carnegie Learning, Inc. 10 Chapter 1 Skills Practice

19 Name Date 9. Given U, and J 1, 5, use a graph to show J U U J. 30. Given Z, 1, use a graph to show Z ( Z ) Carnegie Learning, Inc. 1 Chapter 1 Skills Practice 107

20 Use the given information to answer each question. Round answers to the nearest tenth, if necessary. 31. An airplane is traveling due north at a speed of 750 miles per hour. The plane encounters a crosswind of 5 miles per hour blowing from the west. Draw a diagram of the situation and calculate the resultant speed and direction of the airplane. B C 750 mph A 5 mph D AC , , The resultant speed of the airplane is about 750. miles per hour slightly northeast Carnegie Learning, Inc. 10 Chapter 1 Skills Practice

21 Name Date 3. An airplane is traveling due south at a speed of 00 miles per hour. The plane encounters a crosswind of 0 miles per hour blowing from the east. Draw a diagram of the situation and calculate the resultant speed and direction of the airplane. 010 Carnegie Learning, Inc. 1 Chapter 1 Skills Practice 109

22 33. A boat is traveling 7 meters per second due north toward an island when it encounters a current traveling meters per second due east. Draw a diagram of the situation and calculate the measure of the angle in which the boat is heading toward the island. 3. A boat is traveling across a 00-foot river at feet per second. How much time will it take the boat to cross the river? Carnegie Learning, Inc Chapter 1 Skills Practice

23 Name Date 35. An airplane travels at a speed of 0 miles per hour due east. If the airplane encounters a headwind of 0 miles per hour due west, calculate the resultant velocity and direction of the airplane. Draw a diagram of the situation. 3. An airplane travels 150 miles per hour due east. It encounters a crosswind of 30 miles per hour due north. Calculate the resultant velocity and direction of the airplane. Draw a diagram of the situation. 010 Carnegie Learning, Inc. 1 Chapter 1 Skills Practice 1031

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25 Skills Practice Skills Practice for Lesson 1.3 Name Date SOS or Is It CQD? Multiplying Vectors by Scalars Vocabulary Define the term in your own words. 1. unit vector Problem Set Multiply each vector by the given scalar. Then compare the magnitudes of the two vectors. 1. If S 1,, calculate 3 S. 3 S 3, 1 The magnitude of 3 S is three times the magnitude of S.. If K, 1, calculate K. 010 Carnegie Learning, Inc. 3. If P,, calculate P.. if R ( 0 ), calculate R. 1 Chapter 1 Skills Practice 1033

26 5. if Z 5, 5, calculate Z.. if A ( 3 1 ), calculate 10 A. Use the properties of vectors to show that each statement is true. 7. If U (). U ( 1 3 ), show U ( 1 3 ) 1 ( 3 ) ( 1 ) ( 1 ). If Y,, show 3 Y Y ( 3). 9. If T 5, 0, show 7 T T (7) If 11. If N 1, 1 and R,, show 5( E 3, and F, 1, show ( N R ) 5 N 5 R. E F ) E ( F ). 010 Carnegie Learning, Inc. 103 Chapter 1 Skills Practice

27 Name Date 1. If W ( 0 ) and M ( 3 ), show ( W _ M ) W M. Given V, determine the components of each unit vector u without graphing the vector. 13. V, 3 1. V,0 V 13 u V 1 V 1 13, 3 13, V ( 1 ) 1. V 1, 010 Carnegie Learning, Inc. 17. V, 5 1. V ( 3 ) 1 Chapter 1 Skills Practice 1035

28 Given the magnitude and directional angle of V, use a trigonometric ratio to determine the components of each vector sin 35 y cos sin 35 y 5 cos 35 y.9.1 V.1, Carnegie Learning, Inc. 103 Chapter 1 Skills Practice

29 Name Date Use the given information to answer each question. 5. A boat left the dock at point K and sailed to point R, as shown. When arriving at point R, the crew encountered bad weather and radioed a distress call. Determine the components of the unit vector KL to help the rescue boat locate the sailboat. _ KR 5, K L R 010 Carnegie Learning, Inc. KR 1 _ KL 1 KR KR 1 1 5, 5 1, 1 1 Chapter 1 Skills Practice 1037

30 . An ice skater started skating at point M and skated to point T, as shown. Later, another skater crossed the first skater s path and continued to point H. Determine the components of the unit vector MT to represent the second skater s path. MH 7, H M T 7. A stunt airplane pilot started flying at point S and flew to point B, as shown. The pilot increased his speed at point E. Determine the components of the unit vector SE to represent the location where the pilot increased his speed. _ SB, B E S Carnegie Learning, Inc. 103 Chapter 1 Skills Practice

31 Name Date. A boat sets sail from port and navigates 105 northwest, as shown. If V 7, determine the components of V and determine the components of the corresponding unit vector u. V An airplane takes off and navigates 50 southwest as shown. If V 9, determine the components of V and determine the components of the corresponding unit vector u V 010 Carnegie Learning, Inc. 1 Chapter 1 Skills Practice 1039

32 30. A cruise ship leaves port and navigates 300 southeast as shown. If V 1, determine the components of V and determine the components of the corresponding unit vector u V Carnegie Learning, Inc. 100 Chapter 1 Skills Practice