CE 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 x y z A B

Coordinates of a Point (x, y, and z) A coordinates are (2, 2, 6) B coordinates are (4, -4, -10) x y z B A

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 vector form as: r = x i + y j + z k x y z r A (x,y,z) x i y j z k O

If a position vector extends from point B (x B, y B, z B ) to point A(x A, y A, z A ), then it can be expressed as r BA. By head – to – tail vector addition, we have: r B + r BA = r A then, r BA = r A - r B x y z r BA A (x A,y A,z A ) rArA rBrB B(x B, y B, z B )

Substituting the values of r A and r B, we obtain r BA = (x A i + y A j + z A k) – (x B i + y B j + z B k) = (x A – x B ) i + (y A – y B ) j + (z A – z B ) 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. We know that: F = fu = f ( r / r) x y z r B A F u

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 = f u

Examples Examples 2.12 – 2.15 Problem 86 Problem 98