In three dimensional space we have a plane and a line. These can be oriented in any way. The plane is rotated about the line by n degrees, meaning that originally the position of the plane is fixed to the line and the line is rotated along with it's own axis. How do I form the equation of the rotated plane if I know the equations for the original unrotated plane and the line?
For an example, let's assume the plane is $2\vec{i} + 3\vec{j} - 4\vec{k} + 15 = 0$ and our line is represented by the vector $3\vec{i} - 2\vec{j} + 5\vec{k}$. Let's rotate the plane 25 degrees. How is this done?
Apply Rodrigues' formula to a triple of points $P, Q, R$ on the plane; that'll give you three new points, $P', Q', R'$ and you can compute the new plane as $$ (X - Q') \cdot [(P' - Q') \times (R' - Q')] = 0. $$
As an alternative: First, write your plane equation in the form $$ \newcommand{\vv}{{\mathbf v}} $$ $$ X \cdot \vv + d = 0 $$ In your example, $\vv = [2, 3, -4]$, and $d = 15$, so the equation is $$ X \cdot \begin{bmatrix}2\\3\\4\end{bmatrix} + 15 = 0. $$
Some multiple $c\vv$ of $\vv$ lies on the plane; to see what value of $c$ works, we plug in: $$ (c\vv) \cdot vv + 15 = 0\\ c (\vv \cdot vv) + 15 = 0\\ c (29) + 15 = 0 \\ c = -\frac{15}{29}. $$ In general, let $$ c = -\frac{\vv \cdot \vv}{d}; $$ then the point $c\vv$ lies on the line. Let's call that point $P$.
Now apply Rodrigues' formula, $$ \mathbf{v}_\mathrm{rot} = \mathbf{v} \cos\theta + (\mathbf{k} \times \mathbf{v})\sin\theta + \mathbf{k} (\mathbf{k} \cdot \mathbf{v}) (1 - \cos\theta). $$
to the normal vector $\vv$ of the plane to get a new vector $\vv'$. If we also apply it to the point $P$, we'll get a new point that lies on the rotated plane. But since $P$ is a multiple of $\vv$, $P'$ will be the same multiple of $\vv'$. After a bit of fiddling, the result is that the "d" value remains the same, so
the equation of the new plane is $$ (X \cdot \vv') + d = 0, $$ where $\vv'$ is obtained by Rodrigues' formula, and $d$ is the constant term in the original plane equation.