Another interpretation of the metric tensor, also considered by Gauss, is that it provides a way in which to compute the length of tangent vectors to the surface, as well as the angle between two tangent vectors. In contemporary terms, the metric tensor allows one to compute the dot product of tangent vectors in a manner independent of the parametric description of the surface. Any tangent vector at a point of the parametric surface can be written in the form

for suitable real numbers and . If two tangent vectors are given:

then using the bilinearity of the dot product,

This is plainly a function of the four variables , , , and . It is more profitably viewed, however, as a function that takes a pair of arguments and which are vectors in the -plane. That is, put

This is a symmetric function in and , meaning that

It is also bilinear, meaning that it is linear in each variable and separately. That is,

for any vectors , , , and in the plane, and any real numbers and .

In particular, the length of a tangent vector is given by

and the angle between two vectors and is calculated by

Metric in a vector bundle
দৈর্ঘ এবং কোণ
Inverse metric
Definition