I have a point $(-24.75665066,0.61535793,34.60714434)\in\mathbb{R}^3$
I would like to find the new coordinate after a $3\times3$ rotation matrix is applied to the axis, around the origin. Matrix example below: $$ \begin{bmatrix} -0.4197673 & 0.5603373 & 0.7140151\\ -0.8973154 & -0.1379305 & -0.4192854\\ -0.1364568 & -0.8166990 & 0.5606980 \end{bmatrix} $$
This type of maths is way above my head, however trying to solve this problem is really encouraging me to go and study further mathematics. I was able to find some similar questions, but none I was able to use. Any insight would be much appreciated.
If $A$ is an $m\times n$ matrix, then $f(x)=A\vec{x}$ where $\vec{x}\in\mathbb{R}^n$ describes a linear map $\mathbb{R}^n\rightarrow\mathbb{R}^m$. From your description, it seems that you are working with a $3\times 3$ rotation matrix, which sends vectors (or your "coordinate") in $\mathbb{R}^3\rightarrow\mathbb{R}^3$. Therefore, to apply the transformation to your point, simply express the point as a $3\times 1$ vector, and right-multiply it to the rotation matrix: $$ \begin{bmatrix} -0.4197673 & 0.5603373 & 0.7140151\\ -0.8973154 & -0.1379305 & -0.4192854\\ -0.1364568 & -0.8166990 & 0.5606980 \end{bmatrix} \times \begin{bmatrix} -24.75665066 \\ 0.61535793\\ 34.60714434\\ \end{bmatrix} $$ to obtain the new vector $$ \begin{bmatrix} 35.446864032260734\\ 7.619376905219662\\ 22.279807738857738 \end{bmatrix} $$