Geometric meaning of different dot products So, the standard dot product is defined by the $I$ matrix. If using this dot product between two vectors gives me the result $0$ i can say the vectors are perpendicular because $0$ is the relative lenght of the projection of the first vector on the second one.
If i have a different dot product defined by another matrix, what is the change in his geometric menaning?
 A: You speak of length and perpendicularity (=angle), but those concepts are defined only after a scalar product is given. Geometrically, assigning a scalar product on a vector space amounts to specifying which bases are orthonormal. Once one has an orthonormal base, one can measure lengths and angles. 
The "standard dot product" on $\mathbb R^3$ prescribes that the "standard base" 
$$
(1,0,0), (0,1,0), (0,0,1)$$ 
is orthonormal. But that is only a choice. If you choose a different scalar product, that base is not going to be orthonormal.
A: If you define an inner product with a symmetric matrix $M$ as $u\cdot v=\langle u,v\rangle=u^\top M v$ then that inner product is your description for the shape of the space, in the sense that it tells you how the space is stretched and sheared -- which vectors are orthogonal and so on. From $M,$ one can find an orthonormal basis.
This concept is particularly useful for example in geometry where one may want to study some space and represents points in that space by saying that there is a continuous function from some subset of some Euclidean space to that space. Then you can translate the shape of that space backwards into the Euclidean space by determining some $M$ for each point of that space so that distances and such in the space being studied can be calculated by integrating over a Euclidean space.
In general relativity, there is something like this (not quite an inner product, but a bilinear form as there are some vectors with negative "magnitude" (defined $\langle u,u\rangle$) and some non-zero vectors with zero "magnitude"). This is given by a symmetric matrix which varies in space (and time) which represents how space is stretched and is itself affected by gravity.
A: Suppose we have drawn the two vectors so that their tails are at the same point. The angle between the two vectors has been labelled $\theta$\
$a.b=|a||b|\cos(\theta)$ 
$|a| , |b| $ is the modulus
$\theta$ is the angle between $a$ and $b.$
we sometimes refer to the scalar
product as the dot product.
If  the scalar product of
two vectors which are at right-angles is always 0.
We say that such vectors are perpendicular or orthogonal
