- Used to define a vector.
- Computes the dot product of two vectors.
- Computes the cross product of two vectors.
- Evaluates expressions involving vectors.
- Computes the norm, or length of a vector. For reasons explained below, the use of this commmand is not recommended. A better alternative for our purposes is to use the square root of the dot product of a vector with itself.

> with(linalg): > a := [2, 13, -6]; > b := [5, -4, 17]; > a+b; > 5*a-2*b; > dotprod(a,b); > crossprod(a,b); > crossprod(b,a); > dotprod(a,crossprod(a,b));The next two commands show two different ways to compute the length of a vector. The first way uses the

> norm(a,2); > sqrt(dotprod(a,a));The final example for fixed vectors shows two methods for computing the vector projection or component of

> a_unit := evalm(a/sqrt(dotprod(a,a))); > comp_a1 := evalm(dotprod(b,a_unit)*a_unit);The second method uses the formula for the component.

> comp_a2 := evalm(dotprod(b,a)/dotprod(a,a)*a);In the next set of examples two arbitrary three-dimensional vectors,

> u := [u1,u2,u3]; > v := [v1,v2,v3]; > norm(u,2); > sqrt(dotprod(u,u));The above 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

> assume(u1,real,u2,real,u3,real,v1,real,v2,real,v3,real);The next few commands show how to prove the 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. (Note: the tilda simply means that the computer refered to the assumption, so b1=b1).

> eq1 := dotprod(evalm(u-v),evalm(u-v)); > expand(eq1); > eq2 := dotprod(u,u)+dotprod(v,v)-2*dotprod(u,v); > simplify(eq1-eq2);

- Given the vectors
compute the following:
- a)
- b)

- Show that the following identity is true for any two vectors in three dimensions.

- Given the following triangle
with the points
,, and
- a)
- Show that the triangle is a right triangle using the dot product.
- b)
- Find the area of the triangle using the cross product.

2005-10-31