Need help with Lines, Planes, Curves and Trajectories, Vectors ,algebra homework help

Need help with the files I uploaded for homework.

Would like if possible the best formula to work out the problems and step by step how you solved them.

Q1 and Q2 are related to the Intro file.

You are to create a mathematical model of the motion of the fire pot as it is swung in a circle by the
goblin. Your model must describe the trajectory of the fire pot in the camera frame of reference:
that is, provide and equation for the displacement of the fire pot, as a function of time t, the initial
conditions of the motion and the rotation speed, w (in revolutions per second), relative to the origin
of the camera frame.
Consider the diagram below, which shows quantities relevant to the mathematical description of the
problem.
nxù
û
(t)
fo
d
1) Create a coordinate system aligned to the plane of rotation.
a. Define the coordinates of the center of rotation, C, such that is between 1 and 1.5
meters above the ground and located anywhere in the XY plane).
b. Define a unit normal vector to the plane of rotation, în.
C. Determine a vector û that lies in the plane of rotation, that is both orthogonal to în
and parallel to the ground plane of the world.
d. Obtain a third unit vector ûl, that is orthogonal to both î and û, such that you have
a right-handed orthonormal basis {n,û, û } aligned to the plane.
2) Determine a counter-clockwise parameterisation of a circle of radius r in your plane.
a. Determine a parametric equation (in parameter t) describing the angular position 8
of point P (the position of the firepot) relative to the reference line û. This equation
should involve the initial angular displacement at t = 0 (i.e., 6.) and the angular
speed w, given in revolutions per second.
b. Using the basis vectors {ù,ūl}, define the equation of the circle as a vector-valued
function in the plane (which should involve angle 6).
C. Substitute your parametric equation for 8 (t) to obtain coordinates of P in the plane,
as a function of initial conditions r, 60, w and time t.
d. Use your coordinates in the plane to state a vector parametric equation for r(t), in
the basis {n, û û1}
3) Obtain a vector-valued equation for the displacement of the firepot, in the camera frame of
reference
a. Define either a set of equations, or a transformation matrix, to convert vectors from
the basis {n, û, û } to the external basis {i, j, k} of the world frame
b. Define either a set of equations, or a transformation matrix, to convert vectors from
the basis {i, j, k} to the basis {f, r, d}, being the forward, right and down
orthonormal basis vectors of the camera frame.
C. Define the position of the camera in the world frame and use this to determine a
vector c from the position of the camera to the center of rotation, C.
d. Convert both vectors c and r(t) into the camera frame and add them together, to
obtain the equation for the displacement of the point P from the origin of the
camera frame.
Question 2:
a) Obtain an equation for the trajectory of the firepot, from the moment it is released until it
would strike the ground, in the world frame of reference. This will require you to determine
the initial conditions for the trajectory (position and velocity in the world frame) using your
model developed in Part 1, and then use these to solve for the constants of integration
when developing the trajectory equation.
b) Using your trajectory model, determine the following quantities as functions of the initial
conditions (given as variable quantities):
1. maximum height reached by the firepot
2. maximum range of the firepot from the point of release
c) Determine the rotation speed and time t of release, such that your model from Part 1 would
generate an initial position and velocity that would have the firepot acheive a height of 3
meters at a distance of at least 20 meters from the launch point.