P01-140 – Position Vector in 3D

Position Vector in Three Dimensions

In this part, we expand from our one-dimensional knowledge the idea of a position vector for a particle as it moves in the $xy$-plane or through space. In generalize, we adapt the previous definition to two-dimensional or three-dimensional motion by packaging them into vectors with two or three components respectively. These vectors acts as ”containers” in which each direction of space is described by a component and allow for a very compact way of keeping track of the overall position and motion of the particle.

Position vector in a 2D coordinate system:
Shown below is a particle $M$ in a two-dimensional coordinate system along with its position vector $\overrightarrow{r}$.

In a two-dimensional coordinate system, the position of the particle $M$ is fully described by knowing its coordinates $\left(x_M,y_M\right)$. Its position vector $\overrightarrow{r}=\langle x_M,y_M\rangle $ is the vector that point from the origin $O$ to the particle $M$ and has components

\boxed{\overrightarrow{r}=\langle x_M,y_M\rangle =x_M\hat{x}+y_M\hat{y}}

As the particle moves in the $xy$-plane, its coordinates $\left(x_M,y_M\right)$ will change but so long as we know what they are equal to, we are able to locate the particle exactly.

Position vector in a 3D coordinate system:
Shown below is a particle $M$ in a three-dimensional coordinate system along with its position vector $\overrightarrow{r}$.

In a three-dimensional coordinate system, the position of the particle $M$ is fully described by knowing its coordinates $\left(x_M,y_M,z_M\right)$. Its position vector $\overrightarrow{r}=\langle x_M,y_M,z_M\rangle $ is the vector that point from the origin $O$ to the particle $M$ and has components

\boxed{\overrightarrow{r}=\left\langle x_M,y_M,z_M\right\rangle =x_M\hat{x}+y_M\hat{y}+z_M\hat{z}}

As the particle moves in space, its coordinates $\left(x_M,y_M,z_M\right)$ will change but so long as we know what they are equal to, we are able to locate the particle exactly.