1. ME 221Lecture 31 ME 221 Statics Lecture #3 Sections 2.6 – 2.8

2. ME 221Lecture 32 Announcements HW #1 Due Today Quiz #1 - 15 minutes before the end of the lecture HW #2 due Friday 5/28 Ch 2: 23, 29, 32, 37, 47, 50, 61, 82, 105, 113 Ch 3: 1, 8, 11, 25, 35 Quiz #2 on Monday, May 24 Exam # 1 will be on Wednesday, June 2

3. ME 221Lecture 33 Summary of Last Lecture Writing vector components in terms of base vectors Generating a 3-D unit vector from any given vector

4. ME 221Lecture 34 Nonorthogonal Bases; Linear Equations Resolving vectors onto nonorthogonal directions Setting up and solving linear systems of algebraic equations Resolving Vectors into Components Using Angle Notation

5. ME 221Lecture 35 x y z O A   A=A sin  sin  i+A cos  j+A sin  cos  k Resolving vector into components using angle notation AyAy AxAx AzAz

6. ME 221Lecture 36 Vector Components in Nonorthogonal Coordinate System  xx  y-y- 90+  yy x y AvAv AuAu v u A A: Using Trig (law of sines): AxAx AyAy

7. ME 221Lecture 37 Case 1: One Base Vector Known When vector A is known, subtract A from P x y P   A B x y P -A B = P - A B: Using Vector Addition

8. ME 221Lecture 38 Case 2: Two Directions Known Write out unit vectors x y P   P = A + B P = Pcosβ î + Psinβ ĵ

9. ME 221Lecture 39 Write the components of P: and write the vector sum equation. Next, write the x and y component equations: x-components y-components Here, we have two equations in two unknowns, A and B. Solve the equations.

10. ME 221Lecture 310 For example, using the numerical values: P = 100 lb,  = 10º,  = 20º Set up the system of equations to solve: P cos β = 93.97 = A + 0.866 B x-components P sin β = 34.20 = 0 A + 0.5 B y-components Solving yields: B = 68.4 lb and A = 34.7 lb

11. ME 221Lecture 311 Linear Algebraic Systems Write the x- and y-component equations in matrix form as follows: Solve with your calculator.

12. ME 221Lecture 312 Summary Be able to resolve a vector onto non- orthogonal directions Write the matrix form of the x-, y-, and z- component equationsWrite the matrix form of the x-, y-, and z- component equations Be able to solve a 2 x 2 and 3 x 3 system of equations on your calculatorBe able to solve a 2 x 2 and 3 x 3 system of equations on your calculator

13. ME 221Lecture 313 Multiplying Vectors Section 2.8 There are three basic ways vectors are multiplied –Scalar times a vector –Scalar product –Cross or vector product Often called the “dot” product

14. ME 221Lecture 314 Dot Product Consider two vectors A and B with included angle   A B By definition, the dot product is A B = |A| |B| cos 

15. ME 221Lecture 315 Dot Product of Base Vectors Let A and B be the base vectors and we find Also note that since  = 0, then cos  = 1 since  = 90°, then cos  = 0 ··· ···

16. ME 221Lecture 316 Writing the Components The dot product between two vectors is: Components of a vector may be easily found And finally ·... ·.

17. ME 221Lecture 317 Applications Determine the angle between two arbitrary vectors Components of a vector parallel and perpendicular to a specific direction · · · ||

18. ME 221Lecture 318 Example Problem

19. ME 221Lecture 319 Quiz #1