ES 1022y Engineering Statics Force Vectors
In this course forces (acting along the vector axis), moments (rotation about the vector
axis), and position vectors (moving along the vector axis) are all vector quantities. A
force, moment or position vector has:
A vector quantity can be represented graphically by an arrow that shows its magnitude,
direction and sense.
Magnitude: characterized by size in some units, e.g. 34 N; represented by length of the
arrow according to some scale, say, 1 cm = 10 N → 3.4 cm = 34 N.
Direction: the angle between a reference axis and the arrow's line of action.
Sense: indicated by the arrowhead (one of two possible directions)
A Word on Vector Notation
In the lecture notes and text book a vector quantity is indicated by a letter in boldface
type (F), while the magnitude of a vector is denoted by an italicized letter (F). For
handwritten work a vector is usually indicated by drawing an arrow above the letter
representing the vector, thus
Similarly, unit vectors can be denoted in handwritten work by drawing a hat symbol
above the letter to give
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Consider two force vectors A and B. We want to add them together to find the vector
sum, or resultant force vector, R such that
We can do this using one of two methods.
If the two forces A and B are represented by the adjacent sides of a parallelogram, then
the diagonal of the parallelogram is equal to the vector sum of the two forces.
Triangle of forces:
Special case of the parallelogram law.
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As a special case, if the two vectors A and B are collinear, that is
the parallelogram law reduces to an algebraic or scalar addition
Where we have three or more forces we can either use repeated applications of the
parallelogram law or a force polygon to find the resultant force.
We can also use trigonometry to add two force vectors together using the sine and cosine
laws. Consider a triangle with sides of length A, B, and C, and corresponding interior
angles a, b, and c.
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Resolution of a Force Vector
A force vector can be resolved into two components with known lines of action using the
In cases where we need to determine the resultant of more than two forces it is often
easier to resolve each force into its components along specified axes, before adding these
components algebraically to find the resultant. In this case we usually resolve each force
into components using a Cartesian coordinate system.
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The force F = 450 lb acts on the frame. Resolve this force into components acting along
members AB and AC, and determine the magnitude of each component.
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For a vector A with a magnitude of A the unit vector is defined as
The unit vector has:
The vector A can then be represented as
Cartesian Unit Vectors
In a three-dimensional, rectangular Cartesian axis system the Cartesian unit vectors i, j,
and k are used to designate the directions of the x, y, and z axes respectively.
The Cartesian unit vectors have a dimensionless magnitude of 1 and a sense that is given
by either a plus or minus sign to show whether they are pointing along the positive or
negative x, y or z axes.
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Two-Dimensional or Coplanar Forces
Now consider a two-dimensional, or co-planar, force vector F, as shown.
Resolving the force into components acting along the x and y axes allows us to write the
force F as
The magnitudes of each component of F are represented by the positive scalars F xand F y
These are often referred to as the rectangular components of F.
If we define θ as the angle between the line of action of the force F and the positive x
axis, then we can write
Force Addition Using Component s
Consider three coplanar forces (forces lying in the same plane) F , F 1 an2 F . 3
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These can be represented using Cartesian vector notation as
The resultant force vector is then given by
In the general case we can write
It is important to remember to take sign conventions into account. Components along
positive coordinate axes have positive values and vice versa.
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The magnitude of the resultant force vector is given by
while the angle is given by
Three-Dimensional Force Vectors
In many situations we need to solve problems in three dimensions, rather than the two
that we have considered so far. To do this we use a three-dimensional, rectangular
Cartesian coordinate system that is said to be right handed.
Consider a vector A in 3-D space. .
The vector can be represented as
with a magnitude given by
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The orientation of A is defined by the coordinate direction angles α, β, and γ. These are
measured between the tail of A and the positive x, y, and z axes respectively, and will
always be between 0 and 180 .
The angles are defined by the direction cosines
Recall that in Cartesian vector form A can be written as
The unit vector in the direction of A is then
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This implies that the unit vector can be rewritten in terms of the direction cosines
Recall that the magnitude of a vector is obtained as
Therefore the magnitude of the unit vector is given by
we can also write A in terms of the coordinate direction angles
If one of the coordinate direction angles is missing, we can always work out what it is by
using the equation
and rearranging it to find the cosine