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

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1 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 for example) and, then Directed Quantities: Quantities such as velocity and acceleration (quantities that involve magnitude as well as direction) Vector: A line segment with an assigned direction. Example u =, represents vector u with initial point A, and terminal point B. NOTE: Vectors are denoted with bold letters such as u in this example. Magnitude (or length): denoted by, representing the length of vector found by, where and can be Unit Vectors: A vector with length (magnitude) of 1. Two specific unit vectors are listed i and j. NOTE: ANY VECTOR WITH MAGNITUDE EQUAL TO 1 IS A UNIT VECTOR (not just these two special cases) i = and j = 2. Horizontal & Vertical Components: Given vector, ; is the horizontal component and is the vertical component 3. Component Form of a Vector: If a vector v is represented in the plane with initial point and terminal point, then u + u - cu =, where c 5. Properties of Vectors: See Pg 595 in your text book (Basic operations for addition, subtraction and scalar multiplication also applies to vectors) 6. Vectors in Terms of i and j: The vector can be expressed in terms of i and j by as: = ai + bj 7. Horizontal and Vertical Components of a Vector: Let v be a a vector with magnitude and direction. Then = ai + bj, where and 8. Velocity (as a vector): The velocity of a moving object can be modeled by a vector whose direction is the direction of motion of the object and whose magnitude is the speed of the object. 9. Resultant Force: If several forces are action on an object, the sum of these forces is called the resultant force.

2 10. Example 1 (A): Use the graph to find the vector with initial point P and terminal point Q. Choose the correct answer from the following: a. b. c. d. 14. Example 1 (E): Express the vector with initial point P and terminal point Q in component form. 11. Example 1 (B): Find the vector with initial point P (6, 8) and terminal point Q (4, 4). a. b. c. d. 12. Example 1 (C): Find the vector with initial point P (4, 3) and terminal point Q ( 9, 4). 15. Example 2 (A): Find the magnitude of with initial point and terminal point. 13. Example 1 (D): Express the vector with initial point P and terminal point Q in component form.

3 16. Example 2 (B): Find the magnitude of initial point and terminal point. with 21. Example 3 (D): Given vectors a = and b =, find. 17. Example 2 (C): Find the magnitude of with initial point 22. Example 4 (A): Write the vector u in terms of the vectors i and j given that u = Select the correct answer (rounded to the nearest tenth): 18. Example 3 (A): For vectors u = <3, 4>, < 1, 3>, find u + v. a. u + <2, 7> b. u + <1, 4> c. u + <4, 1> d. u + <7, 2> 23. Example 4 (B): Write the vector u in terms of the vectors i and j given that u = Select the correct answer (rounded to the nearest tenth): 19. Example 3 (B): Given that u =<6, 3>, < 2, 1>, find 7 u 4 v. 20. Example 3 (C): Find 2u, 3v,u + v, and 3u 4v for the given vectors u and v. u =, (a) 2u = 24. Example 4 (C): Find 2u, 3v, u + v, and 3u 4v for the given vectors u and v. u = 5i, 9i 5j (a) 2u = (b) 3 (c) u + (d) 3u 4 (b) 3 (c) u Example 4 (D): Given that u = i + j, i j, find 9u 2v. (d) 3u Example 5 (A): Part I: Find the horizontal of

4 Part II: Find the horizontal and vertical components of the vector v, where v = 34,. Write the vector in the form. a. b. c. d. 31. Example 5 (F): Find the magnitude and direction (in degrees) of the vector. i + j (a) v = (b) = 27. Example 5 (B): Find the horizontal and vertical components of the vector v: v = 3, and write the vector in terms of the vectors i and j. 28. Example 5 (C): Find the direction (in degrees) of the vector < 5, 3 >. a. 34 b. 30 c. 59 d Example 6 (A): A jet is flying through a wind that is blowing with a speed of 40 mi/h in the direction N 35º E. The jet has a speed of 714 mi/h in still air, and the pilot heads the jet in the direction N 50º E. Find the true speed of the jet. 33. Example 6 (B): A pilot heads his jet due east. The jet has a speed of 400 mi/h in still air. The wind is blowing due north with a speed of 37 mi/h. Find the true velocity of the jet as a vector. 29. Example 5 (D): Find the magnitude of the vector v = i + j. a. v = b. v = 4 c. v = d. v = Example 5 (E): Find the magnitude and direction (in degrees) of the vector. If necessary round your answer to two decimal places. (a) v = (b) = 34. Example 6 (C): A pilot heads his jet due east. The jet has a speed of 400 mi/h in still air. The wind is blowing due north with a speed of 41 mi/h. Find the true velocity of the jet as a vector. 35. Example 6 (D): A jet is flying in a direction N 20 E with a speed of 550 mi/h. Find the north and east components of the velocity. (a) The north component is mi/h. (b) The east component is mi/h. (c) Find the true speed of the jet. (d) Find the jet speed if there is a 40 mi/hr. wind in the N35 E direction.

5 36. Example 6 (E): A river flows due south at 5 mi/h. A swimmer attempting to cross the river heads due east swimming at 3 mi/h relative to the water. Find the true velocity of the swimmer as a vector. 38. Example 7 (A): A straight river flows east at a speed of 7 mi/h. A boater starts at the south shore of the river and heads in a direction a from the shore (see the figure). The motorboat has a speed of 20 mi/h in still water. The boater wants to arrive at a point on the north shore of the river directly opposite the starting point. In what direction should the boat be headed? i + j 37. Example 6 (F): A boat heads in the direction N 72 E. The speed of the boat relative to the water is 27 mi/h. The water is flowing directly south. It is observed that the true direction of the boat is directly east. (a) Express the velocity of the boat relative to the water as a vector in component form. cos, sin (b) Find the speed of the water. Round your answer to the nearest tenth. mi/h (c) Find the true speed of the boat. Round your answer to the nearest tenth. mi/h 39. Example 7 (B): A straight river flows east at a speed of 8 mi/h. A boater starts at the south shore of the river and heads in a direction from the shore (see the figure). The motorboat has a speed of 17 mi/h in still water. The boater wants to arrive at a point on the north shore of the river directly opposite the starting point. In what direction should the boat be headed?

40. Example 7 (C): A straight river flows east at a speed of 7 mi/h. A boater starts at the south shore of the river and heads in a direction from the shore (see the figure).

6 40. Example 7 (C): A straight river flows east at a speed of 7 mi/h. A boater starts at the south shore of the river and heads in a direction from the shore (see the figure). The motorboat has a speed of 23 mi/h in still water. The boater wants to arrive at a point on the north shore of the river directly opposite the starting point. In what direction should the boat be headed? 41. Example 8 (A): The forces F 1, F 2,..., F n acting at the same point P are said to be in equilibrium if the resultant force is zero, that is, if F 1 + F F n = 0. F + F + F = i + j (b) Find the additional force required for the forces to be in equilibrium. F = i + j 44. A 100-lb weight hangs from a string as shown in the figure. Find the tensions T and T in the string. F 1 = <3, 1>, F 2 = <8, 5> What additional force F is required for the forces acting at P to be in equilibrium? a. F = < 8, 1> b. F = < 11, 6> c. F = < 3, 6> d. F = < 3, 5> Round your answers to the nearest tenth. T = i + j, 42. Example 8 (B): The forces F 1, F 2,..., F n acting at the same point P are said to be in equilibrium if the resultant force is zero, that is, if F 1 + F F n = 0. F 1 = i + j, F 2 = i j, F 3 = 9 i + 2 j Find the additional force required for the forces to be in equilibrium.p. T = i + j 43. Example 8 (C): The forces F, F...F acting at the same point P are said to be in equilibrium if the resultant force is zero, that is, if F + F +...+F = 0. F = 3i 3j, F = 3i + 3j, F = 6i + 3j (a) Find the resultant forces acting at P. 45. A 200 lb weight hangs from a string as shown in the figure. Find the tensions T 1 and T 2 in the string. 46. The forces F, F...F acting at the same point P are said to be in equilibrium if the resultant force is zero, that is, if F + F F = 0.

7 (a) Find the resultant forces acting at P. Round your answers to two decimal places. F 1 + F 2 + F 3 (b) Find the additional force required for the forces to be in equilibrium. Round your answers to two decimal places. F 4 =

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,