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angle this vector makes with the horizontal. The vector has a length of approximately 5.2 centimeters and is at an approximate angle of 188° with the horizontal. Mid-Chapter Quiz: Lessons 8-1 through 8-3 Find the resultant of each pair of vectors using either the triangle or parallelogram method. State the magnitude of the resultant in centimeters and its direction relative to the horizontal. 3. SOLUTION: Translate s so that its tail touches the tip of p. Then draw the resultant vector p + s as shown. Draw the horizontal. 1. SOLUTION: Drawing may not be to scale. Translate j so that its tail touches the tip of h. Then draw the resultant vector h + j as shown. Draw the horizontal. Measure the length of p + s and then measure the angle this vector makes with the horizontal. The vector has a length of approximately 1.2 centimeters and is at an approximate angle of 330° with the horizontal. Drawing may not be to scale. Measure the length of h + j and then measure the angle this vector makes with the horizontal. The vector has a length of approximately 1.2 centimeters and is at an approximate angle of 323° with the horizontal. 2. 4. SOLUTION: Translate b so that its tail touches the tip of a. Then draw the resultant vector a + b as shown. Draw the horizontal. Drawing may not be to scale. SOLUTION: Translate g so that its tail touches the tip of f. Then draw the resultant vector f + g as shown. Draw the horizontal. Drawing may not be to scale. Measure the length of f + g and then measure the angle this vector makes with the horizontal. The vector has a length of approximately 5.2 centimeters and is at an approximate angle of 188° with the horizontal. eSolutions Manual - Powered by Cognero 3. Measure the length of a + b and then measure the angle this vector makes with the horizontal. The vector has a length of approximately 1.8 centimeters and is at an approximate angle of 102° with the horizontal. 5. SLEDDING Alvin pulls a sled through the snow with a force of 50 newtons at an angle of 35° with the horizontal. Find the magnitude of the horizontal and vertical components of the force. SOLUTION: Draw a vector to represent Alvin pulling the sled. The vector can be resolved into a horizontal component x and a vertical component y as shown. Page 1 angle this vector makes with the horizontal. The vector has a length of approximately 1.8 centimeters and is at an approximate angle of 102° with the horizontal. Mid-Chapter Quiz: Lessons 8-1 through 8-3 5. SLEDDING Alvin pulls a sled through the snow with a force of 50 newtons at an angle of 35° with the horizontal. Find the magnitude of the horizontal and vertical components of the force. The magnitude of the horizontal component is about 41.0 newtons and the magnitude of the vertical component is about 28.7 newtons. 6. Draw a vector diagram of c – 3d. SOLUTION: Draw a vector to represent Alvin pulling the sled. The vector can be resolved into a horizontal component x and a vertical component y as shown. SOLUTION: Rewrite the expression as the addition of two vectors: c − 3d = vector The horizontal and vertical components of the vector form a right triangle. Use the sine or cosine ratios to find the magnitude of each component. c + (−3d). To represent c, draw a the length of c in the same direction as c. To represent −3d, draw a vector 3 times as long as d in the opposite direction from d. Then use the triangle method to draw the resultant vector. The magnitude of the horizontal component is about 41.0 newtons and the magnitude of the vertical component is about 28.7 newtons. 6. Draw a vector diagram of c – 3d. Drawings may not be to scale. Let be the vector with the given initial and terminal points. Write as a linear combination of the vectors i and j. 7. B(3, −1), C(4, −7) SOLUTION: First, find the component form of . SOLUTION: Rewrite the expression as the addition of two vectors: c − 3d = vector c + (−3d). To represent c, draw a the length of c in the same direction as c. To represent −3d, draw a vector 3 times as long as d in the opposite direction from d. eSolutions Manual - Powered by Cognero Then rewrite the vector as a linear combination of the standard unit vectors. 8. B(10, −6), C(−8, 2) SOLUTION: First, find the component form of Page 2 . the standard unit vectors. Mid-Chapter Quiz: Lessons 8-1 through 8-3 8. B(10, −6), C(−8, 2) 11. MULTIPLE CHOICE Which of the following is SOLUTION: First, find the component form of . the component form of with initial point A(–5, 3) and terminal point B(2, −1)? A B C D SOLUTION: Then rewrite the vector as a linear combination of the standard unit vectors. 9. B(1, 12), C(−2, −9) The correct answer is B. SOLUTION: First, find the component form of Find the component form. 12. BASKETBALL With time running out in a game, . Rachel runs towards the basket at a speed of 2.5 meters per second and from half-court, launches a shot at a speed of 8 meters per second at an angle of 36° to the horizontal. Then rewrite the vector as a linear combination of the standard unit vectors. 10. B(4, −10), C(4, −10) SOLUTION: First, find the component form of . a. Write the component form of the vectors representing Rachel’s velocity and the path of the ball. b. What is the resultant speed and direction of the shot? SOLUTION: a. Since Rachel is moving straight forward, the component form of her velocity v 1 is . Use Then rewrite the vector as a linear combination of the standard unit vectors. 11. MULTIPLE CHOICE Which of the following is the component form of with initial point A(–5, 3) and terminal point B(2, −1)? A eSolutions B Manual - Powered by Cognero C D the magnitude and direction of the ball’s velocity v 2 to write this vector in component form. The component form of the vector representing Rachel’s velocity is and the component form of the vector representing the path of the ball is Page 3 b. Add the algebraic vectors representing v 1 and v 2 The component form of the vector representing Mid-Chapter Quiz: 8-1 through 8-3 Rachel’s velocity is Lessons and the component form of the vector representing the path of the ball is 14. Q(1, −5), R(−7, 8) SOLUTION: b. Add the algebraic vectors representing v 1 and v 2 First, find the component form. to find the resultant velocity, vector r. Find the magnitude of the resultant. Next, find the magnitude. Substitute x2 − x1 = −8 and y2 − y 1 = 13 into the formula for the magnitude of a vector in the coordinate plane. The speed of the ball is about 10.2 meters per second. Find the resultant direction angle θ. 15. X(−3, −5), Y(2, 5) SOLUTION: First, find the component form. The speed of the ball is about 10.2 meters per second at an angle of about 27.6° with the horizontal. Find the component form and magnitude of the vector with each initial and terminal point. 13. A(−4, 2), B(3, 6) SOLUTION: First, find the component form. Next, find the magnitude. Substitute x2 − x1 = 5 and y2 − y 1 = 10 into the formula for the magnitude of a vector in the coordinate plane. Next, find the magnitude. Substitute x2 − x1 = 7 and y2 − y 1 = 4 into the formula for the magnitude of a vector in the coordinate plane. 16. P(9, −2), S(2, −5) SOLUTION: First, find the component form. 14. Q(1,Manual 8) by Cognero −5), R(−7, eSolutions - Powered SOLUTION: First, find the component form. Page 4 Next, find the magnitude. Substitute x2 − x1 = −7 and y2 − y 1 = −3 into the formula for the magnitude of a Mid-Chapter Quiz: Lessons 8-1 through 8-3 16. P(9, −2), S(2, −5) SOLUTION: 18. u = ,v= SOLUTION: First, find the component form. Next, find the magnitude. Substitute x2 − x1 = −7 and y2 − y 1 = −3 into the formula for the magnitude of a vector in the coordinate plane. Find the angle θ between u and v to the nearest tenth of a degree. 17. u = ,v= 19. u = ,v= SOLUTION: SOLUTION: 20. u = ,v= SOLUTION: 18. u = ,v= SOLUTION: eSolutions Manual - Powered by Cognero Page 5 Mid-Chapter Quiz: Lessons 8-1 through 8-3 20. u = The correct answer is F. Find the dot product of u and v. Then determine if u and v are orthogonal. ,v= SOLUTION: 22. SOLUTION: Since , u and v are not orthogonal. 23. SOLUTION: Since , u and v are not orthogonal. 24. SOLUTION: 21. MULTIPLE CHOICE If u = and w = F −18 G −2 H 15 J 38 ,v= , find (u ⋅ v ) + (w ⋅ v ). , Since , u and v are not orthogonal. 25. SOLUTION: SOLUTION: Since , u and v are orthogonal. 26. WAGON Henry uses a wagon to carry newspapers for his paper route. He is pulling the wagon with a force of 25 newtons at an angle of 30° with the horizontal. The correct answer is F. Find the dot product of u and v. Then determine if u and v are orthogonal. 22. SOLUTION: eSolutions Manual - Powered by Cognero v are not Since , u and 23. orthogonal. a. How much work in joules is Henry doing when he pulls the wagon 150 meters? b. If the handle makes an angle of 40° with the horizontal and he pulls the wagon the same distance with the same force, is Henry doing more or less Page 6 work? Explain your answer. SOLUTION: Mid-Chapter Quiz: Lessons 8-1 through 8-3 , u and v are orthogonal. Since 26. WAGON Henry uses a wagon to carry newspapers for his paper route. He is pulling the wagon with a force of 25 newtons at an angle of 30° with the horizontal. If the handle makes an angle of 40° with the horizontal, Henry is doing about 2872.7 joules of work. Therefore, Henry is doing less work. Find the projection of u onto v. Then write u as the sum of two orthogonal vectors, one of which is the projection of u onto v. 27. u = ,v= SOLUTION: Find the projection of u onto v . a. How much work in joules is Henry doing when he pulls the wagon 150 meters? b. If the handle makes an angle of 40° with the horizontal and he pulls the wagon the same distance with the same force, is Henry doing more or less work? Explain your answer. SOLUTION: a. Use the projection formula for work. The magnitude of the projection of F onto is The magnitude of the directed distance is 150. To write u as the sum of two orthogonal vectors, start by writing u as the sum of two vectors w1 and w2, or u = w1 + w2. Since one of the vectors is the projection of u onto v , let w1 = projvu and solve for Henry is doing about 3247.6 joules of work pulling the wagon. w2. b. Use the projection formula for work. The magnitude of the projection of F onto is The magnitude of the directed distance is 150. Thus, If the handle makes an angle of 40° with the horizontal, Henry is doing about 2872.7 joules of work. Therefore, Henry is doing less work. 28. u = . ,v= SOLUTION: Find the projection of u onto v . Find the projection of u onto v. Then write u as the sum of two orthogonal vectors, one of which is the projection of u onto v. 27. u = ,v= SOLUTION: eSolutions Manual - Powered by Cognero Find the projection of u onto v . Page 7 Thus, . Mid-Chapter Quiz: Lessons 8-1 through 8-3 28. u = Thus, 29. u = ,v= . ,v= SOLUTION: SOLUTION: Find the projection of u onto v . Find the projection of u onto v . To write u as the sum of two orthogonal vectors, start by writing u as the sum of two vectors w1 and To write u as the sum of two orthogonal vectors, start by writing u as the sum of two vectors w1 and w2, or u = w1 + w2. Since one of the vectors is the projection of u onto v , let w1 = projvu and solve for w2, or u = w1 + w2. Since one of the vectors is the projection of u onto v , let w1 = projvu and solve for w2. w2. Thus, Thus, . 30. u = . ,v= SOLUTION: 29. u = ,v= Find the projection of u onto v . SOLUTION: Find the projection of u onto v . eSolutions Manual - Powered by Cognero Page 8 Mid-Chapter Quiz: Lessons 8-1. through 8-3 Thus, 30. u = ,v= SOLUTION: Find the projection of u onto v . To write u as the sum of two orthogonal vectors, start by writing u as the sum of two vectors w1 and w2, or u = w1 + w2. Since one of the vectors is the projection of u onto v , let w1 = projvu and solve for w2. Thus, eSolutions Manual - Powered by Cognero . Page 9