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

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 kO

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:rB + rBA = rA

then,rBA = rA - rB

x

y

z

rBAA (xA,yA,zA)

rArB

B(xB, yB, zB)

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.We know that:F = fu = f ( r / r)

x

y

z

rB

AF

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 BDetermine the unit vector ( u = r / r ) which has the

direction of both r and FDetermine F by combining its magnitude ( f ) and direction

( u )

F = f u

Examples

Examples 2.12 – 2.15 Problem 86 Problem 98