Mechanics 2
Force Resultants in Three Dimensions
Force Expressed as a Cartesian Vector
If a force lies in an octant of an x, y, z frame, it may be resolved into three rectangular components. By successive application of the parallelogram law yield F = F ^{’} + F_{z}, and F^{ ‘}_{ }= F_{x} + F_{y}. Combining these two equations, we have
F = F_{x} + F_{y} + F_{z}
If a set of i, j, k, unit vector is used to define the positive direction of the x, y, z axes. F can be expressed in Cartesian vector form as
F = F_{x} i + F_{y} j +F_{z} k
Here, F_{x }, F_{y }, F_{z} represent the magnitude of each force component and i, j, k specify the components’ direction.

Magnitude of Force
Direction of Force
These numbers are known as the direction cosines of F. Once they have been obtained, the direction angles α, β, γ can then be obtained from the inverse cosines.
Where cos^{2} α + cos^{2} β + cos^{2} γ = 1
Unit vector (u)
The unit vector is dimensionless,
F = F u and
k
And
Position Vectors
This vector is of importance in formulating a Cartesian force vector directed between any two points in space, and it will be used in finding the moment of a force. The position vector r is defined as a fixed vector which locates a point in space to another point,

r_{A} + r = r_{B}
r = r_{B}  r_{A}
r = (x_{B} i + y_{B} j + z_{B} k) (x_{A} i + y_{A} j + z_{A} k)
r = (x_{B } x_{A})i + (y_{B}  y_{A})j + (z_{B}  z_{A})k
Force vector directed along a line
We can formulate F as a Cartesian vector by realizing that it acts in the same direction as the position vector r directed from point A to point B on the cord.

Thus common direction is specified by the unit vector
u = r /r
F = F u = F (r/r)
Procedure for analysis
When F is directed along a line which extends from point A to point B, then F can be expressed in Cartesian vector form as follows:
Position
vector
Determine the
position vector r directed from A to B, and compute its magnitude r.
Unit
vector
Determine the unite
vector u = r/r which defines the direction of both r and F.
Force
vector
Determine F by
combining its magnitude F and direction u, i.e., F = F
u.
* Example:
A man pulls on the cord shown in the figure with a force of 300 N. Represent this force, acting on the support A, as a Cartesian vector and determine its direction.

Position
vector
The coordinates of the end points of the cord are A(0, 0, 10), B(4, 3, 2). The
position vector from A to B is:
r = (4 – 0)i + (3 – 0)j + (2 – 10)k
= {4i  3 j  8 k} m
The magnitude of r, which represents the length of cord AB, is
Unit vector
Force vector
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The coordinate direction angles are measured between r (or F) and the positive axes of localized coordinate system with origin placed at A. From the components of the unite vector.
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