Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
CE 201 - Statics Lecture 5 Contents Position Vectors Force Vector Directed along a Line POSITION VECTORS If a force is acting between two points, then the use of position vector will help in representing the force in the form of Cartesian vector. As discussed earlier, the right-handed coordinate system will be used throughout the course z B y A x Coordinates of a Point (x, y, and z) A coordinates are (2, 2, 6) B coordinates are (4, -4, -10) z A y x B Position Vectors Position vector is a fixed vector that locates a point relative to another point. If the position vector ( r ) is extending from the point of origin ( O ) to point ( A ) with x, y, and z coordinates, then it can be expressed in Cartesian z vector form as: A (x,y,z) r r=xi+yj+zk zk O xi yj x y If a position vector extends from point B (xB, yB, zB) to point A(xA, yA, zA), then it can be expressed as rBA. By head – to – tail vector addition, we have: z rB + rBA = rA A (x ,y ,z ) r B(x , y , z ) then, r r y rBA = rA - rB B B B B x A BA A A A Substituting the values of rA and rB, we obtain rBA = (xA i + yA j + zA k) – (xB i + yB j + zB k) = (xA – xB) i + (yA – yB) j + (zA – zB) k So, position vector can be formed by subtracting the coordinates of the tail from those of the head. FORCE VECTOR DIRECTED ALONG A LINE If force F is directed along the AB, then it can be expressed as a Cartesian vector, knowing that it has the same direction as the position vector ( r ) which is directed from A to B. The direction can be expressed using the unit vector (u) u = (r / r) where, ( r ) is the vector and ( r ) is its magnitude. z We know that: F = fu = f ( r / r) A F x B r u y Procedure for Analysis When F is directed along the line AB (from A to B), then F can be expressed as a Cartesian vector in the following way: Determine the position vector ( r ) directed from A to B Determine the unit vector ( u = r / r ) which has the direction of both r and F Determine F by combining its magnitude ( f ) and direction (u) F=fu Examples Examples 2.12 – 2.15 Problem 86 Problem 98