9/5/2012PHY 113 A Fall Lecture 41 PHY 113 A General Physics I 9-9:50 AM MWF Olin 101 Plan for Lecture 4: Chapter 3 – Vectors 1.Abstract notion of vectors 2.Displacement vectors 3.Other examples

9/5/2012PHY 113 A Fall Lecture 42

9/5/2012 PHY 113 A Fall Lecture 43

9/5/2012PHY 113 A Fall Lecture 44 iclicker question A.Have you attended a tutoring session yet? B.Have you attended a lab session yet? C.Have you attended both tutoring and lab sessions?

9/5/2012PHY 113 A Fall Lecture 45 Mathematics Review -- Appendix B Serwey & Jewett iclicker question A.Have you used this appendix? B.Have you used the appendix, and find it helpful? C.Have you used the appendix, but find it unhelpful? iclicker question Have you changed your webassign password yet? A.yes B.no

9/5/2012PHY 113 A Fall Lecture 46 Mathematics Review -- Appendix B Serwey & Jewett a b c 

9/5/2012PHY 113 A Fall Lecture 47 Definition of a vector 1.A vector is defined by its length and direction. 2.Addition, subtraction, and two forms of multiplication can be defined 3.In practice, we can use trigonometry or component analysis for quantitative work involving vectors. 4.Abstract vectors are useful in physics and mathematics.

9/5/2012PHY 113 A Fall Lecture 48 Vector addition: a b a – b Vector subtraction: a -b a + b

9/5/2012PHY 113 A Fall Lecture 49 Some useful trigonometric relations (see Appendix B of your text)  c b  a  Law of cosines: a 2 = b 2 + c 2 - 2bc cos  b 2 = c 2 + a 2 - 2ca cos  c 2 = a 2 + b 2 - 2ab cos  Law of sines:

9/5/2012PHY 113 A Fall Lecture 410 Vector components: axax ayay

9/5/2012PHY 113 A Fall Lecture 411 axax ayay   a = 1 m Suppose you are given the length of the vector a as shown. How can you find the components? A.a x =a cos  a y =a sin  B.a x =a sin  a y =a cos  C.Neither of these D.Both of these

9/5/2012PHY 113 A Fall Lecture 412 Vector components; using trigonometry An orthogonal coordinate system A

9/5/2012PHY 113 A Fall Lecture 413 Vector components: axax ayay byby bxbx

9/5/2012PHY 113 A Fall Lecture 414 Examples VectorsScalars Position rTime t Velocity vMass m Acceleration aVolume V Force FDensity m/V Momentum pVector components

9/5/2012PHY 113 A Fall Lecture 415 Vector components Vector multiplication “Dot” product “Cross” product

9/5/2012PHY 113 A Fall Lecture 416 Example of vector addition: a b a + b

9/5/2012PHY 113 A Fall Lecture 417 a b a + b 

9/5/2012PHY 113 A Fall Lecture 418 Webassign version:  Note: In this case the angle  is actually measured as north of east.

9/5/2012PHY 113 A Fall Lecture 419 Another example: