I am trying to understand dot and cross products from a physics perspective. If a space curve r(t) satisfies the equation ${r}^{\prime}(t)\times {r}^{\u2033}(t)=0$ for all t, I understand that the derivative of r′(t) is parallel to r′(t), so the velocity vector r′(t) does not change direction. Thus, this curve moves along a line.

However, I'm not sure what ${r}^{\prime}(t)\cdot {r}^{\u2033}(t)=0$ means. My intuition is that it represents motion along a circle, or part of a circle, since the velocity and acceleration vectors should be perpendicular in that case.

However, I'm not sure what ${r}^{\prime}(t)\cdot {r}^{\u2033}(t)=0$ means. My intuition is that it represents motion along a circle, or part of a circle, since the velocity and acceleration vectors should be perpendicular in that case.