Understanding the use of Pythagorean theorem for the length and square of a vector and how this is the dot product of a vector with itself is no problem. I'm trying to look inside the meaning of the dot product of two different vectors and understand it. I can also accept (just following the dot product definition) the use or meaning of cosine of the angle between two vectors and derivation for this as part of their dot product; but I just cannot understand the meaning of the dot product of two different vectors in a clear way. I have already looked in some YouTube videos and seen a couple of wikipedia articles. Is this just supposed to be unintuitive, and we must just accept it? v*v, okay no trouble. u*w, not understood, only can blindly follow the definition.