Download Position Vectors, Force along a Line

CE 201 - Statics
Lecture 5
Contents
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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
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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
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Examples 2.12 – 2.15
Problem 86
Problem 98