Two Forms of the Dot Product

In ordinary Euclidean space, the dot product of two vectors is the product of the length of both vectors times the cosine of the angle between them.

the inner product, or “dot product,” between the two vectors:

Functions and vectors are kinds of tensors, but to think productively about curved spacetime we’re going to need some more elaborate versions.

When you study vector spaces, you also study the concepts of linear maps, kernel, rank, dimension, and codimension. Vector spaces are usually represented by letters and linear maps by arrows connecting these letters. But when I chose to picture vector spaces as bigger or smaller barrels (according to their dimension) and linear maps as bigger or sm

A vector is defined in mathematics as a quantity that can only be fully described by both a magnitude and a direction.

A vector is just a tensor with one index.

Use two vectors to define a loop, parallel-transport a third vector around the loop, and a fourth vector representing the change in the third will be a measure of how curved the space is.