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

2. 2 Rev.S08 Learning Objectives Upon completing this module, you should be able to: 1. Learn and apply basic concepts about vectors. 2. Perform operations on vectors. 3. Represent a vector quantity algebraically and find unit vectors. 4. Compute dot products and the angle between two vectors. 5. Use vectors to solve applications. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

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

4. 4 Rev.S08 Quick Review on Parallel Lines and Transversal http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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. m n parallel lines q Transversal

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

6. 6 Rev.S08 Basic Terminology http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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.

7. 7 Rev.S08 Basic Terminology Continued http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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 component of vector u. The positive angle between the x-axis and a position vector is the direction angle for the vector.

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

9. 9 Rev.S08 Example of Finding Magnitude and Direction Angle http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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.

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

11. 11 Rev.S08 Example of Finding the Horizontal and Vertical Components http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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.

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

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

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

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

16. 16 Rev.S08 How to Compute Dot Product? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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

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

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

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

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

21. 21 Rev.S08 Example http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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. 10 50 10  v

22. 22 Rev.S08 Example http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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. v u 45 20 

23. 23 Rev.S08 Example http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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?

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

25. 25 Rev.S08 Example Continued http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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 . 10.9

26. 26 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 u is the vector -u. Rev.S08 What is the Equilibrant? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

27. 27 Rev.S08 What have we learned? We have learned to: 1. Learn and apply basic concepts about vectors. 2. Perform operations on vectors. 3. Represent a vector quantity algebraically and find unit vectors. 4. Compute dot products and the angle between two vectors. 5. Use vectors to solve applications. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

28. 28 Rev.S08 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.