Points, Vectors, and Functions
We have one more detail to discuss about vectors: dimensionality.
Vectors can be more than 2-dimensional. A 3-dimensional vector has x-, y-, and z-components:
More generally, an n-dimensional vector has n different components:
(x1, x2, ..., xn).
Finding the direction of such a vector is a little more complicated than in the 2-D case. However, finding the magnitude isn't any more complicated: we square all the components and add them up. The magnitude of the n-dimensional vector (x1, x2, ..., xn) is
Vectors can have any positive integer number of dimensions. Though the notation can get hairy when using letter-symbols for denoting vectors with more than 26 dimensions, those vectors still exist. They are used by all the time in complex problems solved by computer algorithms. It's also possible to have infinite-dimensional vectors.