I understand there is a post on this already. But it doesn't answer my question.
I took two random vectors , $(2,3), (3,1)$. sketched them.
Their dot product is $9 = (2 \times 3 + 3 \times 1)$
What does this number mean?, now apparently its the projection of one vector on another, that makes sense when i think of two vectors pointing in the same direction as they are superimposed you would expect the answer to be the magnitude squared of the vector (by pythag).
I understand vector projection and the formula that is used to determine what component of a vector is pointing in the direction of the another. That is not my question. My question is what is 9 in this case, I had a look on wikipedia and it explains it in terms of $a . b = |a|\cdot|b|\cdot\cos\theta $.
I can derive this fine. But don't understand what it MEANS.
I'm sorry if this is a duplicate, I was unsure whether someone would reply to a comment after a sustained period of time.
Best Answer
Write $a \cdot b = ( |a|) ( |b| \cos \theta)$
Then consider
The projection of B onto A is of length $|b| \cos \theta$, it is the portion of the green line up to the dotted line perpendicular to it (or I suppose beyond it). Then the dot product is the just magnitude of these two vectors, $a$ and $b\cos \theta$, multiplied.
Note: I have interchanged $a$ and $A$, $b$ and $B$ due to the diagram and your question.
Have a look in the preview of this book, Section 1.2 has a good explanation of the dot product:
http://www.amazon.co.uk/dp/3540761802/ref=pd_lpo_sbs_dp_ss_2?pf_rd_p=569136327&pf_rd_s=lpo-top-stripe&pf_rd_t=201&pf_rd_i=0716749920&pf_rd_m=A3P5ROKL5A1OLE&pf_rd_r=0WF1Z8FVX29Z404FW8ST