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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

Engineering 36. Chp 4: Force Resultants (2). Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu. The SCALAR Product or DOT Product Between Two Vectors P and Q Is Defined As. Scalar (Dot) Product of 2 Vectors. Scalar Product Math Properties

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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

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  1. Engineering 36 Chp 4: ForceResultants (2) Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

  2. The SCALAR Product or DOT Product Between TwoVectors P and Q Is Defined As Scalar (Dot) Product of 2 Vectors • Scalar Product Math Properties • ARE Commutative • ARE Distributive • Are NOT Associative • Undefined as (P•Q) is NO LONGER a Vector

  3. Scalar Products With Cartesian Unit Components Thus Scalar Product – Cartesian Comps

  4. Angle Between Two Vectors Scalar Product - Applications • Projection Of A Vector OnA Given Line • For Any Axis Defined By A Unit Vector

  5. Vector Magnitude by DOT • A vector DOTed with itself reveals the Square of the Phythagorean Length • Thus the Vector Magnitude • This is IDEAL forMATLAB >> Pv = [-7 3 11] % [Px*iPy*j Pz*k] Pv = -7 3 11 >> Pm = sqrt(dot(Pv,Pv)) Pm = 13.3791

  6. DOT-Prod Application Summary • Given Two intersecting Vectors or Lines • Parallel & PerpendicularComponents • Given Vector VAB, and lineAC find the || & ┴ Components of VAB, VAD & VDB, relative to line AC

  7. DOT-Prod Application Summary • First Calc θ by method ofthe previous slide • Then Simply Use Trigon Right-Triangle ADB

  8. Example: P2-120 by MATLAB • Determine the magnitudes of the components of F= 600N acting along and perpendicular to segment DE of the pipe assembly • Notes • The Angle θbetween DE & EB (the direction of F) appears to be OBTUSE • Fpar • Fperp

  9. Example: P2-120 by MATLAB % Bruce Mayer, PE % ENGR36 * 18Jul2 % ENGR36_parNperp_Projection_H13e_P2_120_1207.m % % Magnitude of a vector by ANON fcn MagV = @(z) sqrt(dot(z,z)) % % Find unit vector along EB, the Force Direction EBv = [-4 -3 2] % in m => [delX*idelY*j delZ*k] EVm = MagV(EBv) uEB = EBv/EVm % % Find unit Vector along Pipe Segment DE DEv = [0 3 0] DEm = MagV(DEv) uDE = DEv/DEm % % Angle between the unit vectors Q = acosd(dot(uEB,uDE))% in ° % Fm = 600 % in Newtons % % the PARALLEL projection of F on DE Fpar = Fm*cosd(Q) % the PERPENDICULAR projection of F on DE Fperp = Fm*sind(Q) % disp(' ') disp('======================================') disp('Chk by finding F against ED (the opposite of DE)') % Find unit Vector along Pipe Segment DE EDv = [0 -3 0] EDm = MagV(EDv) uED = EDv/EDm % Qchk = acosd(dot(uEB,uED))% in ° FparChk = Fm*cosd(Qchk) FperpChk = Fm*sind(Qchk) Q = 123.8545 Fpar = -334.2516 Fperp = 498.2729 ==================================== Chk by finding F against ED (the opposite of DE) Qchk = 56.1455 FparChk = 334.2516 FperpChk = 498.2729

  10. WhiteBoard Work 1 Let’s WorkSome “Angle”Problems 4 2 3

  11. TBC = 5.3 kN

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