A force, for example, may be given in terms of its magnitude F, its sense
of direction, and its line of action. Such a force can be expressed in vector
form using the coordinates of any two points on its line of action.
The vector sought is

The method is to ?nd n on the line of points

where

Scalar Product of Two Vectors. Angles and Projections of Vectors

The scalar product, or dot product, of two concurrent vectors A and B
is de?ned by

A . B = ABcos?                            (1.2.5)

where A and B are the magnitudes of the vectors and ? is the angle between
them. Some useful expressions are

The projection F? of a vector F on an arbitrary line of interest is determined
by placing a unit vector n on that line of interest, so that

Equilibrium of a Particle

A particle is in  equilibrium when the resultant of all forces acting on it is
zero. In such cases the algebraic summation of rectangular scalar components
of forces is valid and convenient:

Free-Body Diagrams

Unknown forces may be determined readily if a body is in equilibrium and
can be modeled as a particle.The method involves free-body diagrams,
which are simple representations of the actual bodies. The appropriate
model is imagined to be isolated from all other bodies, with the signi?cant
effects of other bodies shown as force vectors on the free-body diagram.

Example 1

A mast has three guy wires. The initial tension in each wire is planned to
be 200 lb. Determine whether this is feasible to hold the mast vertical
(Figure 1.2.6).

FIGURE 1.2.6 A mast with guy wires.

Solution.

The three tensions of known magnitude (200 lb) must be written as vectors.

The resultant of the three tensions is

There is a horizontal resultant of 31.9 lb at A, so the mast would not
remain vertical.

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Bela I. Sandor

Vectors. Equilibrium of Particles. Free-Body Diagrams

Two kinds of quantities are used in engineering mechanics. A scalar quantity has
only magnitude (mass, time, temperature, ). A vector quantity has magnitude
and direction (force, velocity, …). Vectors are represented here by arrows
and bold-face symbols, and are used in analysis according to universally
applicable rules that facilitate calculations in a variety of problems. The vector
methods are indispensable in three-dimensional mechanics analyses, but in simple
cases equivalent scalar calculations are suf?cient.

Vector Components and Resultants. Parallelogram Law

A given  vector F may be replaced by two or three other  vectors that have
the same net effect and representation. This is illustrated for the chosen directions
m and n for the components of F in two dimensions (Figure 1.2.1). Conversely,
two concurrent vectors F and P of the same units may be combined to get
a resultant R

FIGURE 1.2.1 Addition of concurrent vectors F and P.

FIGURE 1.2.2 Addition of concurrent, coplanar vectors A, B, and C.

Any set of components of a vector F must satisfy the parallelogram law.
According to Figure 1.2.1, the law of sines and law of cosines may
be useful.

Any number of concurrent vectors may be summed, mathematically or
graphically, and in any order, using the above concepts as illustrated
in Figure 1.2.3.

FIGURE 1.2.3 Addition of concurrent, coplanar vectors A, B, and C.

Unit Vectors

Mathematical manipulations of vectors are greatly  facilitated by the use
of unit vectors. A unit vector n has a magnitude of unity and a de?ned
direction. The most useful of these are the unit coordinate vectors i, j,
and k as shown in Figure 1.2.4.

FIGURE 1.2.4 Unit  vectors in Cartesian coordinates (the same  i,
j, and k set applies in a parallel x?y?z? system of axes).

The three-dimensional components and associated quantities of a vector F
are shown in Figure 1.2.5. The unit vector n is collinear with F.

FIGURE 1.2.5 Three-dimensional components of a vector F.

The vector F is written in terms of its scalar components and the unit
coordinate vectors,

where

The unit vector notation is convenient for the summation of concurrent
vectors in terms of scalar or vector components: Scalar components
of the resultant R:

Vector components:

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All solid materials deform when forces are applied to them, but often it is
reasonable to model components and structures as rigid bodies, at least
in the early part of the analysis. The forces on a rigid body are generally
not concurrent at the center of mass of the body, which cannot be
modeled as a particle if the force system tends to cause a rotation of the
body.

Moment of a Force

The turning effect of a force on a body is called the moment of the force,
or torque. The moment MA of a force F about a point A is de?ned
as a scalar quantity

M_A = Fd            (1.2.7)

where d (the moment arm or lever arm) is the nearest distance from A
to the line of action of F. This nearest distance may be dif?cult to
determine in a three-dimensional scalar analysis; a vector method
is needed in that case.

Equivalent Forces

Sometimes the equivalence of two forces must be established for
simplifying the solution of a problem. The necessary and suf?cient
conditions for the equivalence of forces F  and F? are that they have the
same magnitude, direction, line of action, and moment on a given
rigid body in static equilibrium. Thus,

F = F’  and M_A = M’_A

For example, the ball joint A in Figure 1.2.7 experiences the same
moment whether the vertical force is pushing or pulling downward
on the yoke pin.

FIGURE 1.2.7 Schematic of testing a ball joint of a car.

Vector Product of Two Vectors

A powerful method of vector mechanics is available for solving complex
problems, such as the moment of a force in three dimensions. The
vector product (or cross product) of two concurrent vectors A and
B is de?ned as the vector V = A ? B with the following properties:
1. V is perpendicular to the plane of vectors A and B.
2. The sense of V is given by the right-hand rule (Figure 1.2.8).
3. The magnitude of V is V = AB sin?, where ? is the angle between A and B.
4. A ? B ? B ? A, but A ? B = (B ? A).
5. For three vectors, A ? (B + C) = A ? B + A ? C.

FIGURE 1.2.8 Right-hand rule for vector products.

The vector product is calculated using a determinant,

Moment of a Force about a Point

The vector product is very useful in determining the moment of a force F
about an arbitrary point O. The vector de?nition of moment is

M_{0 ^{= r x F}}

where r is the position vector from point O to any point on the line of action
of F. A double arrow is often used to denote a moment vector in graphics.
The moment MO may have three scalar components, Mx, My, Mz, which
represent the turning effect of the force F about the corresponding
coordinate axes. In other words, a single force has only one moment about
a given point, but this moment may have up to three components with
respect to a coordinate system,

M_o = M_xi + M_yj + M_zk

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