I have performed numerous calculations of dot products throughout my math courses, but I think I lack a fundamental understanding of what it actually means, beyond the abstract way I have been taught to deal with them. I know the definitions (it's the inner product, or the projection of A on to B), but the answer you get with a dot product, does it have a geometric representation? What specifically does it mean? For example, if you have two vectors (1,4) and (5,0), the dot product is 5. But what does that 5 really mean and can you represent it geometrically. I know it is a scalar and not a vector, but I am hoping there is a way to represent it. Obviously, if you had (2,4) and (5,0) instead, the dot would be twice as large because the angle between them is less and the vectors "share" more in common (a larger first component). So what I am asking is if there is any true meaning to these dot product numbers of 5 and 10 and is there a way to geometrically represent them? Or, are they merely numbers for calculating the angle in between vectors, where a larger dot product means a smaller angle.