1.     find the distance of point P from the line AB whith      A (1,2,0).     B (3,0,-2)

d = abs((P-B)^(A-B))/(A-B)

this formula is applicable in any dimension  

Resp.              d=1.633

2.    find the distance between two lines, say   AB and    CD

E = (A-B)^(C-D)

d = reject(A-C),E)          orthogonal rejection of A-C outside the plane E = (A-B)^(C-D)

this formula is applicable in any dimension`

3.   find the angle ABC with   A (5,9,0).   B(2,3,0)   C(8,3,0)

angle = |<log((A-B)/(C-B))>2|         with <A,2>  =  grade2(A)

Cl(3)  

A=5e1+9e2.    B=2e1+3e2    C=8e1+3e2

q=gp((A-B),inverse(C-B))

angle=magnitude(grade(log1(q),2))

resp.                     angle=1.107

4.   find a rotationn sending a unit vector x to the unit vector y.  

y1 = u x1/u      with  u = sqrt(x1/y1)

Cl(3)

x=e1-e2-e3.    y=e1+e2-e3

x1=normalize(x).     y1=normalize(y)

u=sqrt1(gp(x1,inverse(y1)))

y1=gp(u,gp(x1,inverse(u)))

5.