ME 221Lecture 31 ME 221 Statics Lecture #3 Sections 2.6 – 2.8
ME 221Lecture 32 Announcements HW #1 Due Today Quiz # 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
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
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
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
ME 221Lecture 36 Vector Components in Nonorthogonal Coordinate System xx y-y- 90+ yy x y AvAv AuAu v u A A: Using Trig (law of sines): AxAx AyAy
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
ME 221Lecture 38 Case 2: Two Directions Known Write out unit vectors x y P P = A + B P = Pcosβ î + Psinβ ĵ
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.
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 = = A B x-components P sin β = = 0 A B y-components Solving yields: B = 68.4 lb and A = 34.7 lb
ME 221Lecture 311 Linear Algebraic Systems Write the x- and y-component equations in matrix form as follows: Solve with your calculator.
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
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
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
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 ··· ···
ME 221Lecture 316 Writing the Components The dot product between two vectors is: Components of a vector may be easily found And finally ·... ·.
ME 221Lecture 317 Applications Determine the angle between two arbitrary vectors Components of a vector parallel and perpendicular to a specific direction · · · ||
ME 221Lecture 318 Example Problem
ME 221Lecture 319 Quiz #1