Definition 2.2.1. Position.
The position of an object describes the location of that object relative to some origin as a vector quantity.
Section 2.2 Position
As described above, a position vector is always measured relative to some origin (a fixed location in space) and using some set of coordinates. In general, position is three-dimensional, but for now, we will stick to two dimensions. As with any vector, position can be written either by giving a magnitude and a direction, or by writing it in terms of components: \(\vec{r}= r_x\hat{x} + r_y \hat{y}\text{.}\) A graphical representation of such a vector is shown below. Various letters can be used for position, including the generic \(\vec{r}\) used above and the generic Cartesian letters \(\vec{x}\text{,}\) \(\vec{y}\text{,}\) and \(\vec{z}\text{.}\)
Note 2.2.2. Coordinates.
You will start by using the Cartesian coordinates that are probably familiar from math: \(x\) along the horizontal axis and \(y\) along the vertical axis (plus \(z\) if you need it!). Although these coordinates are familiar, they are also arbitrary, and you will soon find that physics likes to use other coordinates when it is convenient for a situation. This may mean rotating the Cartesian coordinates so they are tilted, using different coordinates for two different objects in the same context, or even using non-Cartesian coordinates such as circular (polar) coordinates.
Exercises Activities
1. Practice: Position Components.
As with any other vector, a position vector can be broken into components, as shown in the figure above. Use what you know about triangles and vector components to write equations relating \(r_x\) and \(r_y\) to \(r\) (the magnitude of the position) and \(\theta\text{.}\)
Answer.
\begin{equation*}
r_x = r\cos\theta
\end{equation*}
\begin{equation*}
r_y = r\sin\theta
\end{equation*}
2. Practice: State Vectors.
Choose a different state (or country) than Oregon and find a simple map. Mark a few of the major cities or landmarks on the map (at least three).
- Choose a coordinate system and an origin and mark them on your map.
- Draw position vectors for each of the locations you marked on your map.
- Write each position vector in rectangular form like \(\vec{r} = r_x\hat{x} + r_y\hat{y}\text{.}\)
- Look up any appropriate numbers to find a numerical position vector for each point.
The State Vectors exercise above is a good example of a calculation. When you do calculations in physics, follow the steps below.
