how would you calculate this?
A circular cylinder, height $14$, base radius $2$, has the axis of rotation!
What is the equation of the cylinder jacket when the center of the base circle is the origin? Enter a solution with the help of rotation matrices and using quaternions.
I know that a cylinder equation is: $f(x,y,z)=0=x²+y²-5, 0 < z < 10$, but how do i use it in this context?
Here is a way to do it without using quaternions.
Refer to https://en.wikipedia.org/wiki/Rotation_matrix#In_three_dimensions for rotation matrix about an axis and an angle. The axis $\vec{u}$ in your case is $(2,1,2)$. You also need to normalize it. Say the angle is $\theta$.
You can then represent the original $(x,y,z)$ by the new $(x',y',z')$ using the rotation matrix. You should use $-\theta$ since you are doing it reversely. This should give you equation $$x=f(x',y',z',\theta)\\y=g(x',y',z',\theta)\\z=h(x',y',z',\theta)$$
Then use the original equation $$x^2+y^2=4, 0\leq z\leq 14$$
to find the new equations.