9.1.2. Points and vectors

Using the global coordinate system of Fig. 9.2, we can express the position of any point in space using its three components (x, y, z). For example, the position of the shoulder in global coordinates is:

\[\begin{split} ~^\text{global}p_\text{shoulder} = \begin{bmatrix} x_\text{shoulder} \\ y_\text{shoulder} \\ z_\text{shoulder} \end{bmatrix} \end{split}\]

where \(~^\text{global}p_\text{shoulder}\) is read as: Position (\(p\)) of the shoulder expressed in the global coordinate system.

While three components are sufficient to express points and vectors in three dimensions, we normally use four components instead, the fourth being 1 for points and 0 for vectors. Therefore, while we express the position (a point) of the shoulder in global coordinates as:

\[\begin{split} ~^\text{global}p_\text{shoulder} = \begin{bmatrix} x_\text{shoulder} \\ y_\text{shoulder} \\ z_\text{shoulder} \\ 1 \end{bmatrix} \end{split}\]

we would express its velocity (a vector) as:

\[\begin{split} ~^\text{global}\vec{v}_\text{shoulder} = \begin{bmatrix} v_\text{x shoulder} \\ v_\text{y shoulder} \\ v_\text{z shoulder} \\ 0 \end{bmatrix} \end{split}\]