P01-030 – Position Vector

Position Vector

In this section, we introduce fundamental quantities necessary for the description of the motion of a particle while limiting ourselves strictly to one-dimensional motion. The motivation in doing so is to favor a strong conceptual understanding of the quantities and their relationships before generalizing to two-dimensional or three-dimensional motion.

Position vector for one-dimensional motion:
As the motion of the particle is strictly one-dimensional, we choose to label $x$ the axis along which the particle moves and choose an origin and the positive direction as shown in the figure below.

The position of the particle is then fully described by knowing its $x$-coordinates i.e. its $x$ location on the axis. We then define the position vector as the vector that points from the origin $O$ to the particle $M$ and is defined by

\boxed{\overrightarrow{r} = x \ \hat{x}}

Thus, the particle $M_1$ in the figure above has position vector

\overrightarrow{r_1}=3\ \hat{x}

while the particle $M_2$ has position vector

\overrightarrow{r_2}=-4\ \hat{x}

Knowing the components $3$ and $-4$ of each position vector respectively then allows us to know exactly where each particle is at that instant.

Note: $\hat{x}$ and $\hat{i}$ are equivalent notations and are used interchangeably. As a reminder, $\hat{x}$ is a vector of length $1$ (unit vector) that points in the positive $x$ direction.