In the vect1 package, there are a number of useful functions for 
performing vector operations.  Those functions are illustrated here.

First save the package file "vect1.mac" to a folder on your computer.
Use File --> Load Package... (or Ctrl+L) to find and load the package.

load("C:/Documents and Settings/Compaq_Owner/My Documents/Maxima/vect1.mac")$

(1)
The norm() function computes the magnitude of a vector.

u: [1,-2,3];

norm( u );

(2)
The normalize() function normalizes a vector.

v: [4,1,-3];

normalize( v );

float(%);

norm(%);

(3)
The dot() function computes the dot product of two vectors.

dot( [1,2], [4,-3] );

u: [1/2,-3,4/3]; v: [-1,2/5,-6];

dot( u, v );

(4)
The cross() function computes the cross products of two 3D vectors.

u: [1,-2,1]; v: [3,-2,2];

cross( u, v );

cross( v, u );

(5)
The angle() function computes the angle between two vectors.

angle( [2,3], [1,0] );

float(%);

u: [4,2,1]; v: [0,4,-5];

angle( u, v );

float(%*180/%pi);

(6)
The allsimp() function attempts to simplify using a variety of techniques.

cos(1)^2+sin(1)^2;

allsimp(%);

1/sqrt(430);

allsimp(%);

(7)
The nabla() function computes the gradient vector.

nabla( 2*x^2+4*y^4, [x,y] );

nabla( x^3*y^2*z + sin(x*y*z), [x,y,z] );