VECTOR REVIEW

Purpose | Vector Review | Exercises

Commands from the linalg packages

Maple has number useful commands for working with vectors. . This page provides a brief review of some of the most basic vector commands. The commands described on this page come from the Maple linalg package; this package must be loaded before any of its commands can be used. Some examples using these commands are given below. More examples can be found in the Help screens for each command.

innerprod
Computes the dot product (also known as the inner product) of two vectors.
crossprod
Computes the cross product of two vectors.
evalm
Evaluates expressions involving vectors (and matrices).
norm
Computes the norm, or length, of a vector. For reasons explained below, the use of the norm command is not recommended. A better alternative for our purposes is to use the square root of the inner product (dot product) of a vector with itself.

The first set of examples below demonstrates how to compute linear combinations of vectors, dot and cross products, lengths, and vector components for fixed vectors. 

  > with(linalg):


  Warning, the protected names norm and trace have been redefined and unprotected



  > a := [2,13,-6];

a := [2, 13, -6]

  > b := [5,-4,17];

b := [5, -4, 17]

  > a+b;

[7, 9, 11]

  > 5*a-2*b;

[0, 73, -64]

  > innerprod(a,b);

-144

  > crossprod(a,b);

[197, -64, -73]

  > crossprod(b,a);

[-197, 64, 73]

Think about why the following is exactly zero: 

  > innerprod(a,crossprod(a,b));

0

The next two commands show two different ways to compute the length of a vector. The first way uses the norm command. Note the 2 as the second argument of the command. This 2 has to be there, or else Maple uses a different norm than the one we want. The second way, using the fact that || u || = sqrt(u.u), is preferred for reasons given in the examples below dealing with arbitrary vectors. 

  > norm(a,2);

sqrt(209)

  > sqrt(innerprod(a,a));

sqrt(209)

In the next set of examples, two arbitrary three-dimensional vectors, u and v, are defined and then these vectors are used to establish the vector identity

(u - v).(u - v) = u.u - 2 u.v + v.v .

Note that when defining an arbitrary vector, that is, a vector whose components are all variables, you only have to give the dimension of the vector to the vector command. For example, to define a four-dimensional arbitrary vector called w the command would be w:= vector(4);. 

  > u := vector(3);

u := array(1 .. 3, [ ])

  > v := vector(3);

v := array(1 .. 3, [ ])

The next two commands show the two ways of computing the length of an arbitrary vector. Note the absolute value signs appearing in the output of the norm command. These appear because Maple is allowing the components of u to be complex numbers. The problem is that there is not a simple way to tell Maple that the components of u are real numbers. Absolute value signs cause problems in proving vector identities involving real vectors, so you probably are better off using the square root of the dot product of a vector with itself to compute the length. 

  > norm(u,2);

norm(u,2)

  > sqrt(innerprod(u,u));

sqrt(innerprod(u,u))

The next few commands show how to prove the vector identity given above. The strategy is to compute both sides of the equation and then compare them. If they are identical, then the equation holds. 

  > eq1 := innerprod(evalm(u-v),evalm(u-v));

eq1 := u12 - 2u1v1 + v12 + u22 - 2u2v2 + v22 + u32 - 2u3v3 + v32 

  > eq2 := innerprod(u,u)+innerprod(v,v)-2*innerprod(u,v);

eq2 := u12 - 2u1v1 + v12 + u22 - 2u2v2 + v22 + u32 - 2u3v3 + v32 

  > simplify(eq1-eq2);

0

In the above lines, the use of evalm and simplify is optional; the same results would be found if these commands were left out. In some cases, however, these sorts of clarifying commands are required for Maple to obtain the correct answers.

Purpose | Vector Review | Exercises


Written by: JDF (E-Mail: bach@wpi.edu)
Last modified: Monday, 18 August 2003
Copyright 2003, Joseph D. Fehribach