In the following I compute the matrix mrot which shall rotate the vector v to the unit vector pointing into the x-direction. This does indeed work, except that the last printout for verification, view(u*mrot), looks complicated, despite the fact that it should just be (1, 0, 0). How would I let sage reduce the enormous formulas for the vector components?
v1 = var('v1');
v2 = var('v2');
v3 = var('v3');
e1 = vector([1,0,0]);
v = vector([v1, v2, v3]);
u = v/v.norm();
#u = v;
view(["u=", u], sep=" ");
z=u.cross_product(e1);
#z=e1.cross_product(u);
view(["z=", z]);
c=u.dot_product(e1);
view(["c=",c]);
mat=Matrix( [[0, -z[3-1], z[2-1]], [z[3-1], 0, -z[1-1]], [-z[2-1], z[1-1], 0]] );
#view([mat, mat*mat]);
mrot=-(matrix.identity(3) + mat + (1/(1+c)) * mat*mat);
view(u*mrot);
