I've been using rotation matrices for quite some time now without fully grasping them. Whenever I tried to develop an intuitive understanding of...[tex] x' = x\cos\theta - y\sin\theta \\ y' = x\sin\theta + y \cos\theta [/tex]... I failed and gave up. I've looked at numerous online texts and videos, but following the step-by-step explanations didn't lead to me seeing the whole picture as I had hoped. Could someone explain to me (like I'm 5 years old), why [itex]-y\sin\theta[/itex] and [itex]x\sin\theta[/itex] are used to affect the value along the other axis? Looking at the following picture (pardon the quality): Is the contribution of [itex]y[/itex] to [itex]x[/itex] and vice versa there, to ensure that [itex]P[/itex] maintains the correct distance to the origin, or is that a misguided simplification of mine? The yellow line cannot be [itex]sin + cos[/itex] (Pythagorean theorem) yet I may combine these two to get [itex]x'[/itex] and [itex]y'[/itex]. Do you see where my gap in understanding lies? Is there a drawing that could clarify how these terms combine to give the correct value we observe? Algebraic proofs don't work with me I'm afraid, I need a geometric/visual explanation.