ENGR 3340: Fundamentals of Statics and Dynamics Fundamentals of Statics and Dynamics - ENGR 3340 Professor: Dr. Omar E. Meza Castillo Department of Mechanical Engineering

MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 2 Tentative Lectures Schedule TopicLecture Dot Product 5

MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Vectors Topic 5: Dot Product 3 One thing you learn in science is that there is no perfect answer, no perfect measure. A. O. Beckman

MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 2.9 Dot Product. Applications For this geometry, can you determine angles between the pole and the cables? For force F at Point A, what component of it ( F 1 ) acts along the pipe OA? What component ( F 2 ) acts perpendicular to the pipe? 4

MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 2.9 Dot Product. Definition The dot product of vectors A and B is defined as: Angle  is the smallest angle between the two vectors and is always in a range of 0º to 180º. Dot Product Characteristics : 1. The result of the dot product is a scalar (a positive or negative number). 2. The units of the dot product will be the product of the units of the A and B vectors. Laws of Operation : 1. Commutative law: A B = B A 2. Multiplication by an scalar: a( A B ) = (a A ) B = A (a B ) = ( A B )a 3. Distributive law: A ( B+D) = ( A B ) + ( A D ) 5

MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 2.9 Dot Product. Cartesian Vector Formulation For example considering the Cartesian unit vectors: The dot product of two vectors A and B which are expressed in Cartesian vector form is: 6

MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 2.9 Dot Product. Angle Between two Vectors For the given two vectors in the Cartesian form, one can find the angle by : 1. Finding the dot product 2. Finding the magnitudes (A & B) of the vectors A & B, and 3. Using the definition of dot product and solving for , i.e., where 0     180  7

MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 2.9 Dot Product. Projection of a Vector You can determine the components of a vector parallel and perpendicular to a line using the dot product : 1. Find the unit vector, u aa’ along line aa’. 2. Find the scalar projection of A along line aa by: A || = A u aa’ = A x u x + A y u y + A z u z 3. If needed, the projection can be written as a vector, A ||, by using the unit vector u aa’ and the magnitude found in step (2), A || = A || u aa’ 4. The scalar and vector forms of the perpendicular component can easily be obtained by : A  = (A 2 - A || 2 ) ½ A  = A – A || 8

MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 9

MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics Homework3  bayamon.inter.edu/omeza/ bayamon.inter.edu/omeza/ Omar E. Meza Castillo Ph.D. 10

MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 11 ¿Preguntas? Comentarios

MSP21 Universidad Interamericana - Bayamón ENGR 3340: Fundamentals of Statics and Dynamics 12 GRACIAS