What is the trajectory of R=RxA?

Question

I need to calculate the trajectory of the following expression: $$\vec{r}=\vec{r} \times \vec{A}$$, where $\vec{A}$ is a constant vector.

I think it's a straight line, but can't find a unique line. Solving the equation analytically, considering $\vec{A}=A\hat{x}$ without loss of generality, I get two straight lines, while the trajectory should be a unique one.

Any help is appreciated.

Answer 1

$\vec{r} \times A$ is perpendicular to $\vec{r}$; the only way they can be equal is if $\vec{r}$ is zero.

An algebraic proof is

$$\vec{r} \cdot \vec{r} = \vec{r} \cdot (\vec{r} \times A) = A \cdot (\vec{r} \times \vec{r}) = 0$$

and the zero vector is the only vector whose squared norm is zero.

Answer 2

$$\vec{r}=\vec{r}×\vec{A}$$ Suppose $\vec{r},\vec{A}$ lies in x-y plane$$$$ then $\vec{r}×\vec{A}$ lies in z direction, $$$$ If they are equal only if $\vec{r}=0$