Motion of a projectile We all are aware how we can use the forces and vectors to describe a classical projectile motion (No drag forces).
Major things we already know include:

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*Position-time dependence of projectiles

*Range of projectile

*Distance traversed by projectile

*Curvature at any point

My question relates to a 3D-Geometrical interpretation of the motion.
Consider a cubical environment with appropriate dimensions and align the x,y,z axes with the cube.
We start a projectile at one of the bottom corner and throw it such that it reaches the diagonally opposite corner. Take the angle of launch to be $\theta$ and the initial velocity as $u$.
My interests and questions:

*

*I want to find the system of equations of planes which will intersect at a point and give me the coordinates.

*How many varieties of such planes are possible?

 A: That movement is ruled by an acceleration, which is the second derivative of position.
That means that you only need to specify two things: initial position, and initial velocity, (That includes magnitude and direction).
Given the initial position and the velocity, the plane in which the movement occurs is perfectly determined. So you would just choose that plane and work in two dimensions. It is not necessary to complicate it.
However, if you want to work in 3D, you can also do it. Just split your initial position and your initial velocity into its three components and you've got it:
$$ \begin{cases} \displaystyle x=x_0+v_{ox}t+ \frac{1}{2} a_x t^2 \\
y=y_0+v_{oy}t+ \frac{1}{2} a_y t^2 \\
 z=z_0+v_{oz}t+ \frac{1}{2} a_z t^2
\end{cases} $$

Edit:
Now that I see your comment, I understand what you mean.
Of course there are infinite possible moving planes whose intersection is one moving point.
The easiest solution is the one I said, just set:
$$ \begin{cases} \displaystyle x=x_0+v_{ox}t+ \frac{1}{2} a_x t^2 \\
y=y_0+v_{oy}t+ \frac{1}{2} a_y t^2 \\
 z=z_0+v_{oz}t+ \frac{1}{2} a_z t^2
\end{cases} $$
and those are actually three planes. The first one is $x=x_{particle}$, and so on.
Then, since they are planes
Any linear combination of those planes gives you another plane with the same solution.
That's the key of linear equations.
Check that all those planes depend on a parameter: $t$, which coincides with the time, but it is a free parameter after all.
One last thing, check that this question should belong to Mathematics SE
