Understanding Vectors: Operations and Applications

1. MAC 1114 • Module 9 • Introduction to Vectors Rev.S08

2. Learning Objectives • Upon completing this module, you should be able to: • Learn and apply basic concepts about vectors. • Perform operations on vectors. • Represent a vector quantity algebraically and find unit vectors. • Compute dot products and the angle between two vectors. • Use vectors to solve applications. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

3. Introduction to Vectors There are two major topics in this module: - Introduction to Vectors, Operations, and the Dot Products - Application of Vectors http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

4. q Transversal m parallel lines n Quick Review on Parallel Lines and Transversal • Parallel lines are lines that lie in the same plane and do not intersect. • When a line q intersects two parallel lines, q, is called a transversal. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

5. Name Angles Rule q Alternate interior angles 4 and 5 3 and 6 Angles measures are equal. m Alternate exterior angles 1 and 8 2 and 7 Angle measures are equal. n Interior angles on the same side of the transversal 4 and 6 3 and 5 Angle measures add to 180°. Corresponding angles 2 & 6, 1 & 5, 3 & 7, 4 & 8 Angle measures are equal. Important Angle Relationships http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

6. Basic Terminology • A vector in the plane is a directed line segment. • Consider vector AB • A is called the initial point • B is called the terminal point • Magnitude: length of a vector, expressed as • The sum of two vectors is also a vector. • The vector sum A + B is called the resultant. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

7. Basic Terminology Continued • A vector with its initial point at the origin is called a position vector. • A position vector u with its endpoint at the point (a, b) is written • The numbers a and b are the horizontal component and vertical componentof vector u. • The positive angle between the x-axis and a position vector is the direction angle for the vector. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

8. What are Magnitude and Direction Angle of Vector ? • The magnitude (length) of vector u = is given by • The direction angleθ satisfies where a≠ 0. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

9. Example of Finding Magnitude and Direction Angle • Find the magnitude and direction angle for • Magnitude: • Direction Angle: • Vector u has a positive horizontal component. • Vector u has a negative vertical component, placing the vector in quadrant IV. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

10. What are the Horizontal and Vertical Components? • The horizontal and vertical components, respectively, of a vector u having magnitude |u| and direction angle θ are given by • That is, http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

11. Example of Finding the Horizontal and Vertical Components • Vector w has magnitude 35.0 and direction angle 51.2°. Find the horizontal and vertical components. • Therefore, w = • The horizontal component is 21.9, and the vertical component is 27.3. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

12. Example • Write each vector in the Figure on the right in the form http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

13. Solutions http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

14. What are Vector Operations? • For any real numbersa, b, c, d, and k, http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

15. Example: Vector Operations find: and Let • a) 4v • b) 2u + v • c) 2u− 3v http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

16. How to Compute Dot Product? • A unit vector is a vector that has magnitude 1. • Dot Product • The dot product of two vectors • is denoted u • v, read “u dot v,” and given by http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

17. Example of Finding Dot Products • Find each dot product. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

18. What are the Properties of the Dot Product? For all vectorsu, v, and w and real numbers k, • a) • b) • c) • d) • e) • f) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

19. What is the Geometric Interpretation of Dot Product? • If θ is the angle between the two nonzero vectors u and v, where 0°≤θ≤ 180°, then http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

20. Example of Finding the Angle Between the Two Vectors • Find the angleθbetween two vectors • By the geometric http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

21. v 10 10 θ 50 Example • Forces of 10 newtons and 50 newtons act on an object at right angles to each other. Find the magnitude of the resultant and the angle of the resultant makes with the larger force. • The resultant vector, v, has magnitude 51 and make an angle of 11.3° with the larger force. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

22. 45 v 20° u Example • A vector w has a magnitude of 45 and rests on an incline of 20°. Resolve the vector into its horizontal and vertical components. • The horizontal component is 42.3 and the vertical component is 15.4. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

23. Example • A ship leaves port on a bearing of 28.0° and travels 8.20 mi. The ship then turns due east and travels 4.30 mi. How far is the ship from port? What is its bearing from port? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

24. Example Continued • Vectors PA and AE represent the ship’s path. • Magnitude and bearing: http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

25. 10.9 Example Continued • The ship is about 10.9 mi from port. • To find the bearing of the ship from port, find angle APE. • Add 20.4° to 28.0° to find that the bearing is 48.4°. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

26. What is the Equilibrant? • We have learned how to find the resultant of two vectors. • A vector that will counterbalance the resultant is called the equilibrant. For instance, the equilibrant of vector uis the vector -u. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

27. What have we learned? • We have learned to: • Learn and apply basic concepts about vectors. • Perform operations on vectors. • Represent a vector quantity algebraically and find unit vectors. • Compute dot products and the angle between two vectors. • Use vectors to solve applications. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

28. Credit • Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: • Margaret L. Lial, John Hornsby, David I. Schneider, Trigonometry, 8th Edition http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08