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Engineering Statics: Open and Interactive

Section 2.1 Vectors

You can visualize a vector as an arrow pointing in a particular direction. The tip is the pointed end and the tail the trailing end. The tip and tail of a vector define a line of action. A line of action can be thought of as an invisible string along which a vector can slide. Sliding a vector along its line of action does not change its magnitude or its direction. Sliding a vector can be a handy way to simplify vector problems.


Adjust the magnitude and direction of the vector with the dot at the tip, and adjust the point of application by dragging the line of action and sliding the tail along it.
Figure 2.1.1. Vector Definitions
The standard notation for a vector uses the vector’s name in bold font, or an arrow or bar above the vector’s name. All three of these notations mean the same thing.
\begin{equation*} \vec{F} = \vecarrow{F} = \bar{F} = \textrm{ a vector named } F \end{equation*}
Most printed works including this book will use the bold symbol to indicate vectors, but for handwritten work, you and your instructor will use the bar or arrow notation.
Force vectors acting on physical objects have a point of application, which is the point where the force is applied. Other vectors, such as moment vectors, are free vectors, which means that the point of application is not significant. Free vectors can be moved freely to any location as long as the magnitude and direction are maintained.
The vector’s magnitude is a positive real number including units which describes the ‘strength’ or ‘intensity’ of the vector. Graphically a vector’s magnitude is represented by the length of its vector arrow, and symbolically by enclosing the vector’s symbol with vertical bars. This is the same notation as for the absolute value of a number. The absolute value of a number and the magnitude of a vector can both be thought of as a distance from the origin, so the notation is appropriate. By convention the magnitude of a vector is also indicated by the same letter as the vector, but in non-bold font.
\begin{equation*} F = |\vec{F}| = \textrm{ the magnitude of vector }\vec{F} \end{equation*}
By itself, a vector’s magnitude is a scalar quantity, but it makes no sense to speak of a vector with a negative magnitude so vector magnitudes are always positive or zero. Multiplying a vector by -1 produces a vector with the same magnitude but pointing in the opposite direction.
Vector directions are described with respect to a coordinate system. A coordinate system is an arbitrary reference system used to establish the origin and the primary directions. Distances are usually measured from the origin, and directions from a primary or reference direction. You are probably familiar with the Cartesian coordinate system with mutually perpendicular \(x\text{,}\) \(y\) and \(z\) axes and the origin at their intersection point.
Another way of describing a vector’s direction is to specify its orientation and sense. Orientation is the angle the vector’s line of action makes with a specified reference direction, and sense defines the direction the vector points along its line of action. A vector with a positive sense points towards the positive end of the reference axis and vice-versa. A vector representing an object’s weight has a vertical reference direction and a downward sense or negative sense, for example.
A third way to represent a vector is with a unit vector multiplied by a scalar component. Unit vectors are vectors with a magnitude of one (unitless), and scalar components are signed values with units. Together, they fully define a vector quantity; the unit vector specifies the direction of its line of action, and the scalar component specifies its magnitude and sense. The scalar component “scales” the unit vector.
Be careful not to confuse scalar components, which can be positive or negative, with vector magnitudes, which are always positive.
Vectors are either constant or vary as a function of time, position, or something else. For example, if a force varies with time according to the function \(F(t) = (10\ \textrm{N/s}) t\text{,}\) where \(t\) is the time in seconds, then the force will be \(\N{0}\) at \(t=\second{0}\) and increase by \(\N{10}\) each second thereafter.