Skip to content

Cartesian Position

Definition and Properties

Cartesian position vector (Groves, p47) Cartesian position vector (Groves, p47)

Figure 1 Cartesian position vector (Groves, p47)

As shown in Figure 1, Cartesian position of an origin of frame \(F_\alpha\) with respect to an origin of frame \(F_\beta\) resolved about the axes of frame \(F_\gamma\) is denoted as:

\[ \mathbf{r}^{\gamma}_{\beta \alpha} = \left[ \begin{array}{ccc} x^{\gamma}_{\beta \alpha} & y^{\gamma}_{\beta \alpha} & z^{\gamma}_{\beta \alpha} \end{array} \right]^T. \]
Properties
Inverse \(\mathbf{r}^{\gamma}_{\beta \alpha} = -\mathbf{r}^{\gamma}_{\alpha \beta}\)
Addition \(\mathbf{r}^{\gamma}_{\beta \alpha} = \mathbf{r}^{\gamma}_{\delta \alpha} - \mathbf{r}^{\gamma}_{\delta \beta} = \mathbf{r}^{\gamma}_{\beta \delta} + \mathbf{r}^{\gamma}_{\delta \alpha}\)
Resolving Axes Transformation \(\begin{align*}\mathbf{r}^{\gamma}_{\beta \alpha} &= \mathbf{R}^{\gamma}_{\delta} \mathbf{r}^{\delta}_{\beta \alpha} \\ \mathbf{r}^{\beta}_{\beta \alpha} &= \mathbf{R}^{\beta}_{\delta} \left( \mathbf{r}^{\delta}_{\beta \delta} + \mathbf{r}^{\delta}_{\delta \alpha} \right) \\ &= \mathbf{r}^{\beta}_{\beta \delta} + \mathbf{R}^{\beta}_{\delta} \mathbf{r}^{\delta}_{\delta \alpha} \end{align*}\)