If you have a 3D curve defined parametrically by $t$ for example:

$$P=\cos\left(t+\frac{\pi}{4}\right) \frac{3}{2} {\bf i} + 3 \sin\left(t+\frac{\pi}{4}\right) {\bf j} + \cos\left(t+\frac{\pi}{4}\right) \frac{3\sqrt3}{2} {\bf k}$$

How do you determine if it lies in a certain plane?