How do we combine two waves to create Lissajous figures?

[LCSphysicist]

Homework Statement

All below

Relevant Equations

All below

To answer (a), i imagineed the oscillations parallels (say to z), so we simply add ξ1+ξ2
for b, i imagined two vectors ortogonais, representing the oscillations, so we should add √(ξ1² + ξ2²), is this right?

 

[anuttarasammyak]

The particles move on xy plane as
$$\xi_1\ \mathbf{e_x}+\xi_2\ \mathbf{e_y}$$
which draw Lissajous figures.