I'm new to maxima. And I have two questions:
1. Question I have a little problem with computing vector's magnitude. Now I'm using such form as
sqrt(v1.v1)
This code looks very ugly.
2. Question I thought vector cross product is expressed like a~b. but Maxima says ~ is not an infix operator
As far as I know, there is no built-in function to calculate a vector's magnitude, but you can easily define your own:
(%i1) norm(x) := sqrt(x . x)$
(%i2) norm([1, 1]);
(%o2) sqrt(2)
The cross product operator ~ is only available after loading 'vect':
(%i1) load("vect")$
(%i2) [1, 2, 3] ~ [2, 3, 4];
(%o2) [1, 2, 3] ~ [2, 3, 4]
(%i3) express(%);
(%o3) [- 1, 2, - 1]
If you don't mind using lists as vectors a very simple solution would be functions::
cross(u, v) := [u[2] * v[3] - v[2] * u[3], v[1] * u[3] - u[1] * v[3], u[1] * v[2] - v[1] * u[2]]$
dot(u, v) := u[1] * v[1] + u[2] * v[2] + u[3] * v[3]$
norm(u) := sqrt(dot(u, u))$
this is not an ideal nor safe solution but a concise one.
well, I didn't like the vec library, so I decided to write my own functions. here it goes:
/*this function gets 3 values as elements of a 3D vector and gives a (3,1) column matrix as the representation of the vector */
vec(vec_tempvar_a1,vec_tempvar_b1,vec_tempvar_c1):=block(
matrix([vec_tempvar_a1],[vec_tempvar_b1],[vec_tempvar_c1]))$
/*this function gets a (3,1) column matrix as representation of a 3D vector and delivers a scalar as the magnitude of the vector*/
vecmag(vecmag_tempvar_V1):= block(
sqrt(transpose(vecmag_tempvar_V1).vecmag_tempvar_V1))$
dotprod(dotprod_tempvar_V1,dotprod_tempvar_V2):=block(
transpose(dotprod_tempvar_V1).dotprod_tempvar_V2)$
/*this function calculates the vector/cross product of the vector on the left on the vector on the right*/
crosprod(crosprod_temvar_V1,crosprod_temvar_V2):= block(
matrix([(crosprod_temvar_V1[2,1]*crosprod_temvar_V2[3,1]-crosprod_temvar_V1[3,1]*crosprod_temvar_V2[2,1])],
[(crosprod_temvar_V1[3,1]*crosprod_temvar_V2[1,1]-crosprod_temvar_V1[1,1]*crosprod_temvar_V2[3,1])],
[(crosprod_temvar_V1[1,1]*crosprod_temvar_V2[2,1]-crosprod_temvar_V1[2,1]*crosprod_temvar_V2[1,1])])
)$
I have also been working on a maxima library for rigitd body dynamics, specially robotic applications using screw theory in particular. you may find it at github.com/Foadfs/RMaxima.
Here is the code that defines the cross product of (n-1) vectors of R^n:
(%i1) cross([vv]):=block([n,U],
n:length(vv)+1,
U:args(ident(n)),
(-1)^(n+1)determinant(apply(matrix,cons(U,vv)))
)$
(%i2) cross([a,b]);
(%o2) [-b,a]
(%i3) cross([a,b,c],[x,y,z]);
(%o3) [bz-cy,cx-az,ay-bx]
(%i4) cross([a,b,c,d],[x,y,z,w],[α,β,γ,δ]);
(%o4) [-b(zδ-wγ)+c*(yδ-wβ)-d*(yγ-zβ),a*(zδ-wγ)-c*(xδ-wα)+d*(xγ-zα),-a*(yδ-wβ)+b*(xδ-wα)-d*(xβ-yα),a*(yγ-zβ)-b*(xγ-zα)+c*(xβ-yα)]
