The calculator treats a vector as a matrix with only one row.

The vector above looks like [ 1 3 5 ] (no commas) in the history area of the HOME screen.

The vector above looks like [ [ 1, 3, 5 ] ] if you copy it from the history area to the command line.
 

Vectors may have any number of components, or entries; on the calculator you're limited to 999.

You can display vectors with two components in Polar coordinates as well as Rectangular coordinates.
You can display vectors with three components in Cylindrical or Spherical coordinates as well as Rectangular coordinates.
 

The first component of vector v is v[ 1, 1 ]. (First row, first entry.)
The second component of vector v is v[ 1, 2 ]. (First row, second entry.)
So [ 1, 3, 5 ][ 1, 3 ] is 5.
 

The standard vector operations are in the Vector ops submenu of the Matrix submenu of the MATH menu (2nd 5).
You can also find them in the CATALOG or type them in by hand if you remember their official names.

• unitV( applied to a vector produces a unit vector (a vector of length 1) with the same direction. unitV([1,2,-2]) = [1/3,2/3,-2/3].
• crossP( applied to a pair of vectors each with three components gives the cross product of those vectors. (The cross product of two vectors is a vector with length equal to the product of the lengths multiplied by the sine of the angle between them and direction perpendicular to both of them, chosen so that v, w, and v x w form a right-hand system. See your calculus book for details.) crossP([1,3,5],[1,2,-2]) = [-16,7,-1].
• dotP( applied to a pair of vectors gives the dot product of those vectors. (The dot product is a number which is the sum of the products of the corresponding components of the two vectors. Geometrically, it's the product of the lengths of the vectors multiplied by the cosine of the angle between them.) dotP([1,2,3,4],[1,3,5,7]) = 50.
• The next four command convert vectors with two components to Polar format, vectors with two or three components to Rectangular format, and vectors with three components to Cylindrical or Spherical format.
• The command you don't see here is norm(, which computes the length of a vector. It's on the Norms submenu of the Matrix submenu of the MATH menu (2nd 5.) You can see in the screen above that the Norms submenu is item H on the Matrix submenu. You can also find it in the CATALOG or type it in by hand if you remember it's called Norm and not Length. You can also compute the square root of the dot product of the vector with itself.