Cross Product and Moment Vector Formulation
In this informative content, learn about the mathematical operation of cross product on vectors, including its notation and graphical representation. Explore moment vector formulation, Cartesian vector formulation, and examples illustrating the calculation of moments caused by forces. Delve into the concepts with detailed explanations and images.
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Presentation Transcript
Chapter Four By Laith Batarseh Home Previous Next End
MOMENT VECTOR Cross product Cross product is a mathematical operation can be done on vectors Cross product for one time is done for two vectors The cross product of two vectors is a vector perpendicular to the plane of A and B The notation of vector A cross vector B is: C = AxB where C is the resultant vector from the cross product the vector C can be represented as : C =CUc where Uc is a unit vector in a direction perpendicular to the plane that contains both A and B. The value of the scalar quantity Cis given as : C=A.B.sin( ) where is the angle between A and B. Home Previous The cross product is controlled by the right-hand rule Next End
MOMENT VECTOR Graphical representation Uc Home Previous Next End
MOMENT VECTOR Cartesian vector formulation z K=ixj j=kxi y i=jxk Home x Previous Next End
MOMENT VECTOR Cartesian vector formulation A = Axi + Ayj + Azk B = Bxi + Byj + Bzk AxB=(AyBz -Az By)i-(Ax Bz - AzBx)j + (Ax By - AyBx)k i j k x = B A A A A x y z Home B B B Previous x y z Next End
MOMENT VECTOR Moment vector formulation M Mo = rxF d x Magnitude: Mo = rFsin( ) = Fd O r y Direction: perpendicular to x-y plane (z-direction) F i j k Best for three dimensional problems = x = F r Mo r r r Matrix notation: x y z F F F x y z Home Resultant moment:MRo= (rxF) Previous Next End
MOMENT VECTOR Example [1] Find the moment caused by the following forces about point O z F = [5i + 10j + 6k]N O y 3m 1m 2m Home x Previous Next End
MOMENT VECTOR Example [1] 1. Formulate the position vector (r) : r = 3i+2j+1k 2. Find the moment vector (Mo) by matrix notation i 3 j k 1 = = = + Mo r F 2 2 13 20 x i j k 5 10 6 r = 3i+2j+1k F = [5i + 10j + 6k]N Home Previous Next End
MOMENT VECTOR Example [2] Find the moment caused by the following forces about point O z O y x F = [-5i + 5j -5k]N Home Previous Next End
MOMENT VECTOR Example [2] 1. Formulate the position vector (r) : r= 15i + 10j +6k 2. Find the moment vector (Mo) by matrix notation i j k 6 = = = + + Mo r F 15 10 80 45 125 x i j k 5 5 5 Home Previous Next End
MOMENT VECTOR Summary Moment is a vector can be found by cross product and matrix notation Matrix notation: i j k = = Mo r F x r r r x y z F F F x y z Home Previous Next End
