3.1: Vectors in Mechanics
A vector (or Euclidean vector) is a unitless geometric object with a magnitude and direction in three-dimensional space. In science and engineering a vector can also have a unit. In this section we discuss several particular conventions and methods that are important for using and drawing vectors in mechanics.
3.1.1 Position vector
Probably the most important vector in dynamics is the position vector \(\overrightarrow{\boldsymbol{r}}_{A}\) which represents the position of a point or point mass \(A\) in space by a straight arrow that points from the origin of the coordinate system to point \(A\). Usually, a coordinate system is chosen such that the position vector of the origin has zero magnitude, \(\left|\overrightarrow{\boldsymbol{r}}_{O}\right|=0\).
3.1.2 Relative position vector
A relative position vector \(\overrightarrow{\boldsymbol{r}}_{B / A}\) (see Figure 3.1) is a vector that indicates the relative position of a point (or point mass) \(B\) with respect the position of another point \(A\) and is defined as:
\[\overrightarrow{\boldsymbol{r}}_{B / A} \equiv \overrightarrow{\boldsymbol{r}}_{B}-\overrightarrow{\boldsymbol{r}}_{A} \tag{3.1} \label{3.1}\]
Note that the order of the objects in the subscript is important: \(\overrightarrow{\boldsymbol{r}}_{B / A}\) indicates the position of object \(B\) with respect to the position of \(A\), and vice versa \(\overrightarrow{\boldsymbol{r}}_{A / B}\) indicates the position of object \(A\) with respect to the position of \(B\), inverting the direction of the vector, therefore \(\overrightarrow{\boldsymbol{r}}_{B / A}=-\overrightarrow{\boldsymbol{r}}_{A / B}\), as is also seen from Equation 3.1. A way to remember this is to replace the symbol / by the words ’with respect to’. So, \(\overrightarrow{\boldsymbol{r}}_{B / A}\) is ’the position vector of point \(B\) with
respect to point \(A^{\prime}\). A normal position vector (without / in the subscript) is always measured with respect to the origin \(O\) of the coordinate system.
The relative position vector is convenient to determine the shortest distance between two points in space. Relative velocity and acceleration vectors between two points are defined analogously:
\[\begin{align} \overrightarrow{\boldsymbol{v}}_{B / A} & \equiv \overrightarrow{\boldsymbol{v}}_{B}-\overrightarrow{\boldsymbol{v}}_{A} \tag{3.2} \label{3.2}\\ \overrightarrow{\boldsymbol{a}}_{B / A} & \equiv \overrightarrow{\boldsymbol{a}}_{B}-\overrightarrow{\boldsymbol{a}}_{A} \tag{3.3} \label{3.3}\end{align}\]
The distance \(d_{B / A}\) between two points \(A\) and \(B\) in space is given by the absolute value of the relative position vector connecting these two points.
\[d_{B / A} \equiv\left|\overrightarrow{\boldsymbol{r}}_{B / A}\right|=\sqrt{\overrightarrow{\boldsymbol{r}}_{B / A} \cdot \overrightarrow{\boldsymbol{r}}_{B / A}} \tag{3.4} \label{3.4}\]
Determine the distance between points \(A\) and \(B\) using the relative position vectors.
Solution
From Figure 3.1 we find that: \(\overrightarrow{\boldsymbol{r}}_{A}=(8 \hat{\boldsymbol{\imath}}+2 \hat{\jmath}) \mathrm{m}\) and \(\overrightarrow{\boldsymbol{r}}_{B}=(3 \hat{\boldsymbol{\imath}}+4 \hat{\boldsymbol{\jmath}}) \mathrm{m}\). With Eq. \((3.1)\) we find that \(\overrightarrow{\boldsymbol{r}}_{B / A}=([3-8] \hat{\boldsymbol{\imath}}+[4-2] \hat{\boldsymbol{\jmath}}) \mathrm{m}\). Then we obtain from Equation 3.4 that the distance between points \(A\) and \(B\) is: \(d_{B / A}=\sqrt{5^{2}+2^{2}} \mathrm{~m}=\sqrt{29} \mathrm{~m}\).
3.1.3 Vectors and unit vectors
Besides position vectors, we will deal with many other types of vectors for analysing dynamical systems, like velocity, angular velocity, force and torque vectors. In all these cases the vector, for instance a force \(\overrightarrow{\boldsymbol{F}}\), expresses a magnitude and a direction. The magnitude of the vector can always be obtained by taking the absolute value (norm) of the vector:
\[|\overrightarrow{\boldsymbol{F}}|=\sqrt{\overrightarrow{\boldsymbol{F}} \cdot \overrightarrow{\boldsymbol{F}}} \tag{3.5} \label{3.5}\]
The magnitude \(|\overrightarrow{\boldsymbol{F}}|\) is a scalar with the unit newton (N). The direction of the force can be specified by a unit vector \(\hat{\boldsymbol{F}}\), which is a vector with magnitude 1 that points in the same direction as \(\overrightarrow{\boldsymbol{F}}\). That unit vector is defined as:
\[\hat{\boldsymbol{F}} \equiv \frac{\overrightarrow{\boldsymbol{F}}}{|\overrightarrow{\boldsymbol{F}}|} \tag{3.6} \label{3.6}\]
Because they are a ratio of two quantities with the same unit, unit vectors, despite their name, have no unit. They are called ’unit’ vector because they have a magnitude of unity). With these definitions any vector can be written as the product of a magnitude and a unit vector like in this example for the force vector:
\[\overrightarrow{\boldsymbol{F}}=|\overrightarrow{\boldsymbol{F}}| \hat{\boldsymbol{F}} \tag{3.7} \label{3.7}\]