Why change a given basis? Why would we want to transform a vector in our normal basis (xyz axes) to another basis? The only situation I can recall is when we are looking at a force applied on an inclined plane. Are there any other real life examples where this may be necessary?
Let $A$ be an $n\times n$ matrix with eigenvalues $\lambda_1,\dots,\lambda_n$ and corresponding (linearly independent) eigenvectors $v_1,\dots,v_n$. Then for any (positive) integer $k$ and any vector $x$ we have $$A^kx=c_1\lambda_1^kv_1+\cdots+c_n\lambda_n^kv_n$$ where $c_1,\dots,c_n$ are the coordinates of $x$ with respect to the basis $\{{v_1,\dots,v_n\}}$. So if you find yourself in a situation where you are repeatedly multiplying by the matrix $A$ (and that is very common in applications), the calculations become very simple in the new basis.
Whenever you whant to express a given vector as a sum of $n$ components in chousen $n-dimentional$ space. In the example you gave about forces on inclined plane, you try o express the force in $R^{2}$ that acts on the body ass composed of vertical and horizontal components respect to inclined plane of the displacement, but you could choose any other two, (not parallel) directions to express your force.