The cube length of a vector

Question

The form $$ {\bf r} \cdot {\bf r} = \left|{\bf r}\right|^2 = \sum_{i,j} r_i r_j {\bf e}_i \cdot {\bf e}_j = \sum_{i,j} r_i r_j \delta_{ij} = \sum_i r_i r_i $$

is defined via a bilinear form that takes two vectors and outputs a real number with certain conditions. This quantity is usually referred to as the square of a vector. I have encountered here forms such as ${\bf r}^3, {\bf r}^4, ...$. Now ${\bf r} \cdot {\bf r} \equiv {\bf r}^2$ and is $SO(3)$ invariant. How does one show ${\bf r}^3$ is not $SO(3)$ invariant and consequently only even powers are invariant under $SO(3)$.

The question is tantamount to: How is ${\bf r}^j$ ($j>2$) defined?

Answer 1

You want $\mathbf{r}^{j+k}=\mathbf{r}^j\otimes\mathbf{r}^k$ for a suitable definition of $\otimes$, right? The definition$$\mathbf{r}^{2k}=(\mathbf{r}\cdot\mathbf{r})^k,\,\mathbf{r}^{2k+1}=\mathbf{r}^{2k}r$$works (if $\otimes$ multiplies scalars by scalars and vectors in the usual ways, and takes the dot product of two vectors). This is the standard definition of $\mathbf{r}^n$. However, since $\mathbf{r}^{2k}=r^{2k}$ with $r:=|\mathbf{r}|$, $\mathbf{r}^{2k}$ is often simply written as $r^{2k}$.