Edit: as per the comments below, the interpretation discussed in this question only makes sense if both vectors have strictly positive coordinates.
I've been looking at the dot product between two vectors lately and tried to link the formula to the geometric interpretation. The formula reads:
$a•b = |a||b|cos(\theta)$
In two dimensions, say $a =[x_1, y_1]$, and $b =[x_2, y_2]$.
Then, on the right-hand side, $|b|cos(\theta)$ is just the projection of $b$ onto $a$, and this is multiplied by the length of $a$: so this could be interpreted as an area of a rectangle (and yes, if the projection of $b$ onto $a$ is zero, the area will be zero).
The left-hand side would evaluate as $x_1*x_2 + y_1*y_2$.
Interestingly, $x_1$ is the projection of vector "$a$" onto the x-axis, $y_1$ is the projection of vector "$a$" onto the y-axis, $x_2$ is the projection of "$b$" onto the x-axis and $y_2$ is the projection of "$b$" onto the y-axis.
So the left-hand side can again be viewed as a sum of two rectangle-areas (and yes, if any of the projections of the two vectors on any of the two axes will be zero, then the area will be zero).
So really, the dot product is telling us that:
LHS= the area of the rectangle, whose sides are the projections of the two vectors onto the x-axis PLUS the area of the rectangle whose sides are the projections of the two vectors onto the y-axis
is equal to:
RHS: the area of the rectangle whose sides are (i) the first vector's projection onto the second vector and (ii) the length of the second vector.
My question: what is the intuition for this? I.e. when I draw it or try to imagine it, I don't immediately see that the sum of the areas of the first two rectangles should equal the area of the third. Also, how would I prove that this relationship is always true.