I am trying to understand how $(1),(2) \text{ and } (3)$ are valid if the coordinate is a right-handed coordinate system. The definition of a right-handed coordinate system is a one such that $\overrightarrow{i} \times \overrightarrow{j} = \overrightarrow{k}$ is valid. To me, to prove that I must compute all possibilities to verify that result. Is there any other method?
$$\overrightarrow{i} \times \overrightarrow{j} = \overrightarrow{k} \qquad(1)\\ \overrightarrow{j} \times \overrightarrow{k} = \overrightarrow{i} \qquad(2)\\ \overrightarrow{k} \times \overrightarrow{i} = \overrightarrow{j} \qquad(3)$$
I want to very that $(1),(2) \text{ and } (3)$ is valid as long as it is a right-handed coordinate system. what I mean by possibilities is to try all the different ways that $x,y$ and $z$ can be. For example:
[https://en.wikipedia.org/wiki/Cross_product#Computing_the_cross_product]