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Section 3.3 73 Vectors in the Plane Course Number Section 3.3 Vectors in the Plane Instructor Objective: In this lesson you learned how to write the component forms of vectors, perform basic vector operations, and find the direction angles of vectors. Important Vocabulary Date Define each term or concept. Vector v in the plane The set of all directed line segments that are equivalent to given directed line segment PQ, written v = PQ. Standard position The representative of a set of equivalent directed line segments whose initial point is the origin. Zero vector A vector whose initial point and terminal point both lie at the origin; denoted by 0 = 〈0, 0〉. Unit vector A vector v such that ||v|| = 1. Standard unit vectors The unit vectors 〈1, 0〉 and 〈0, 1〉, denoted by i = 〈1, 0〉 and j = 〈0, 1〉, which can be used to represent any vector v = 〈v1, v2〉. Direction angle The angle (measured counterclockwise) that a unit vector makes with the positive x-axis. I. Introduction (Page 291) A directed line segment has an terminal point and a initial point What you should learn How to represent vectors as directed line segments . The magnitude of the directed line segment PQ, denoted by ||PQ|| , is its length . The magnitude of a directed line segment can be found by . . . using the distance formula. II. Component Form of a Vector (Page 292) A vector whose initial point is at the origin (0, 0) can be uniquely represented by the coordinates of its terminal point (v1, v2). This is the component form of a vector v v = v1 , v 2 , where v1 and v2 are the , written components of v. The component form of the vector with initial point P = (p1, p2) and terminal point Q = (q1, q2) is PQ = 〈q1 − p1, q2 − p2〉 = 〈v1, v2〉 = v. Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. All rights reserved. What you should learn How to write the component forms of vectors 74 Chapter 3 Additional Topics in Trigonometry The magnitude (or length) of v is: (q1 − p1)2 + (q2 − p2)2 ||v|| = = v12 + v22 Example 1: Find the component form and magnitude of the vector v that has (1, 7) as its initial point and (4, 3) as its terminal point. v = 〈3, − 4〉; ||v|| = 5 III. Vector Operations (Pages 293−295) In operations with vectors, numbers are usually referred to as . Geometrically, the product of a vector v and scalars a scalar k is . . . the vector that is |k| times as long as the What you should learn How to perform basic vector operations and represent them graphically vector v. If k is positive, kv has the k is negative, kv has the same opposite To add two vectors geometrically, . . . direction as v, and if direction. position them (without changing length or direction) so that the initial point of one coincides with the terminal point of the other. This technique is called the parallelogram law for vector addition because the vector u + v, often called the resultant of vector addition, is . . . the diagonal of a parallelogram having u and v as its adjacent sides. Let u = 〈u1, u2〉 and v = 〈v1, v2〉 be vectors and let k be a scalar (a real number). Then the sum of u and v is the vector: u+v= 〈u1 + v1, u2 + v2〉 and the scalar multiple of k times u is the vector: ku = k〈u1, u2〉 = 〈ku1, ku2〉 Example 2: Let u = 〈1, 6〉 and v = 〈− 4, 2〉. Find: (a) 3u (b) u + v (a) 〈3, 18〉 (b) 〈− 3, 8〉 Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. All rights reserved. Section 3.3 75 Vectors in the Plane Let u, v, and w be vectors and c and d be scalars. Complete the following properties of vector addition and scalar multiplication: 1. u + v = v+u 2. (u + v) + w = 3. u + 0 = u + (v + w) u 4. u + (− u) = 0 5. c(du) = (cd)u 6. (c + d)u = cu + du 7. c(u + v) = cu + cv 8. 1(u) = u 9. 0(u) = 0 10. ||cv|| = |c| ||v|| IV. Unit Vectors (Pages 295−296) To find a unit vector u that has the same direction as a given nonzero vector v, . . . divide v by its magnitude, that is: u = v/||v||. In this case, the vector u is called a direction of v . unit vector in the Example 3: Find a unit vector in the direction of v = 〈− 8, 6〉. v/||v|| = 〈− 8/10, 6/10〉 = 〈− 0.8, 0.6〉 Let v = 〈v1, v2〉. Then the standard unit vectors can be used to represent v as v = called the v2 is called the v1i + v2j , where the scalar v1 is horizontal component of v vertical component of v sum v1i + v2j is called a linear combination and the scalar . The vector of the vectors i and j. Example 4: Let v = 〈− 5, 3〉. Write v as a linear combination of the standard unit vectors i and j. v = − 5i + 3j Example 5: Let v = 3i − 4j and w = 2i + 9j. Find v + w. v + w = 5i + 5j Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. All rights reserved. What you should learn How to write vectors as linear combinations of unit vectors 76 Chapter 3 V. Direction Angles (Page 297) If u is a unit vector and θ is its direction angle, the terminal point Additional Topics in Trigonometry What you should learn How to find the direction angles of vectors of u lies on the unit circle and u = 〈x, y〉 = 〈cos θ, sin θ〉 (cos θ)i + (sin θ)j = Now, if v is any vector that makes an angle θ with the positive x-axis, it has the same direction as u and v= ||v|| 〈cos θ, sin θ〉 = ||v|| (cosθ)i + ||v|| (sin θ)j If v can be written as v = ai + bj, then the direction angle θ for v can be determined from tan θ = b/a . Example 6: Let v = − 4i + 5j. Find the direction angle for v. 128.66° VI. Applications of Vectors (Pages 298−299) Describe several real-life applications of vectors. Answers will vary. What you should learn How to use vectors to model and solve real-life problems Homework Assignment Page(s) Exercises Larson/Hostetler Trigonometry, Sixth Edition Student Success Organizer IAE Copyright © Houghton Mifflin Company. All rights reserved.