Positions and Translations

Now that we have defined coordinate frame let's say r, we can define other constructs like points, vectors, etc with respect to the defined coordinate frame.

Like a point in 3D space be defined as a vector from the origin of the coordinate frame to that point.

$$p_r = \begin{bmatrix} x_r \\ y_r \\ z_r \end{bmatrix}$$

We say that point p's coordinates are expressed as $\begin{bmatrix} x_r \ y_r \ z_r \end{bmatrix} ^ T$ in the coordinate frame r. The x_r, y_r, and z_r are the projections of the point p onto the x, y, and z axes of the coordinate frame r, respectively.

Translation/Displacement between points

We can translation/displacement represent it as:

$$\vec{d} = p_2 - p_1$$