Engineering Statics

Section 2.1 Vectors

Vectors are physical quantities described by a magnitude and a direction. Vectors are usually given a symbolic name which, in this book, will be typeset in bold like \(\vec{r}\) or \(\vec{F}\) to indicate the vector nature of the quantity. Computations involving vectors must always consider the directionality of each term and follow the rules of vector algebra, which will be described in this chapter. The primary vector quantity you will encounter in statics will be force, but moment and position vectors are also important.

We can visualize a vector as an arrow pointing in a particular direction. The tip is the pointed end and the tail the trailing end.

As vectors include direction, they operate along 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 the vector, however, because sliding does not change either the magnitude or the direction. This can be a handy tool to simplify vector problems.

Force vectors have a point of application, which is the point at which the force is applied. Other vectors, such as moment vectors, are free vectors, which means that the point of application is not significant, and they can be moved freely to any location as long as the magnitude and direction is maintained.

The vector’s magnitude is a positive real number including units that describes the ‘strength’ or ‘intensity’ of the vector. Graphically we represent a vector's magnitude by the length of its vector arrow, and symbolically by enclosing the vector by 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 an non-bold font.

Scalars are physical quantities which have no associated direction and can be described by a positive or negative number, or even zero. Scalar quantities follow the usual laws of algebra, and most scalar quantities have units. Mass, time, temperature, and length are all scalars.

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, 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 weight vector has a vertical orientation 90° from the horizontal axis, with a downward sense.

A third way to represent the direction of a vector is with its unit vector. A unit vector is a vector with a length of one which has the same orientation and sense as a given vector. Hence, a unit vector is a pure direction, as it is independent of the magnitude and unit of measurement of the given vector. Unit vectors are discussed in more detail later in Cartesian Unit Vectors.

Later we will introduce scalar components as a means of describing vectors. These are scalar quantities which may be positive or negative which may lead to some confusion.

Vectors can either be either constant or vary as a function of time, position, or something else. For example, if a force varied with time according to the function \(F(t) = 10t\) [N] where \(t\) is the time in seconds, then the force would be \(\N{0}\) at \(t=0\text{,}\) and increase by \(\N{10}\) each second thereafter.