Calc III/Vector Calculus problems (moving particles in space curves)

"Two particles travel along the space curves

and

Do they collide, and do their paths intersect?"

I calculated that and but I'm not entirely sure where to go from there or how I can use this information to solve the problem. I'm guessing I should find a t where all x coordinates and y coordinates and z coordinates are equal for the intersecting part? But how do I determine if they collide?

Edit: I'm also learning about shapes in 3D (hyperboloids, paraboloids, etc.) While I'm pretty good at determining what the shape will be based on the equation, I'm having trouble visualizing and drawing the graphs of difficult ones like hyperbolic paraboloids. Does anyone know of a (free) program that will allow me to graph functions of 2 variables so I can visualize this better? Thanks!