Image I have a point A = (4,4) and direction vector B = (1,0).
I know that DOT(A,B) = the distance from point A to the closest point along vector B. If B is not normalized, I can get the distance by doing DOT(A,B) / Length(B).
However imagine the direction vector B is not normalized. Lets say B = (6,0).
What does DOT(A, B) geometrically represent now. The value of DOT((4,4), (6,0)) = 24 but I don't know what this actually represents.
Best Answer
Define $J(v_1,v_2):=(-v_2,v_1)$, i.e., $Jv$ is the vector $v$ rotated by $\pi/2$. Observe that the dot product of any two vectors $v$ and $w$ equals $\det(v,Jw)$. In words: the dot product of $v$ and $w$ is the orientated area of the parallelogram spanned by $v$ and $Jw$.