diverse
define basis vectors e0, e1, e2, e3, e12, e23, e13, e123
signature = (1,1,1)                        e1² = e2² = e3² =1
define a vector                             a = (0,1,3,-2,0,0,0,0)
define a bivector                           B = (0,0,0,0,3,-1,1,0)
 
geometric product                         a B = (0,-7,1,-2,0,0,0,-10)
inner product                                a . B = (0,-7,1,-2,0,0,0,0)
outer product                               a ^ B = (0,0,0,0,0,0,0,-10)
 
projection a on B                          
rejection a from B                      
return 0 -> proj ⊥ rej
 
B rotation plane
rotate e1-e2 with axis e1+e2+2e3 by 90°
R1 = (0,0.816497, 0.816497,-0.816497 ,0,0,0,0)
R2 = (0,1,-1,0,0,0,0,0) = e1-e2
R2 = (0,1,-1,0,0,0,0,0) = e1-e2
 
reflection e1-e2 on e1+2e2+e3               r1 = 1/3(-4e1+e2-e3)
reflection r1 on e1+2e2+e3                    r2 = e1-e2
 
meet of two planes                               m = -e1-2e2
 
a = e0+2e2-3e13
B = e12+2e13
# inverse : 1/a
                                                 = (1/2,0,2,0,0,0,7/2,1) /15
                                                 = (1,0,0,0,0,0,0,0) = e0
                                                 = (0,0,0,0,-1,0,-2,0) /5
                                                 = (1,0,0,0,0,0,0,0) = e0
 

                                           = (0,1,-2,0,0,0,0,0) = e1-2e2
 

                      = (0,0.2,0.3,0.5,0,0,0,0) = 0.2e1+0.3e2+0.5e3
 
pow (sqr, 2) = (0,1,0,-2,0,0,0,0) = e1-2e3
 

x = x1*e1 + x2*e2 + x3*e3
xx a scalar function                      (x1²+x2²+x3²,0,0,0,0,0,0,0)
result in the xx gradient                (0,2x1,2x2,2x3,0,0,0,0)
a vector function               (0,x1*sin(x2),x2*cos(x1),0,0,0,0,0)
nabla(xx2,x) result in a scalar function and a bivector function (cos(x1)+sin(x2),0,0,0,x1*cos(x2)+x2*sin(x1),0,0,0)
 
nabla²(xx2,x) the Laplacian: (0,-x1*sin(x2),-x2*cos(x1),0,0,0,0,0)
Cl(3,0)
 
a = e1 + 3e2 - 2e3  
B = 3e12 - e23 +e13  
 
gp(a,B)  
inp(a,B)  
outp(a,B)  
                                 
proj = project(a,B)  
rej = reject(a,B)  
inp(proj, rej)  
                                 
B=dual(e1+e2+2e3)
R1=RR(e1-e2,B,90)  
 
R2=RR(R1,e1+e2+2e3,-90)  
R2=RR(R1,e1+e2+2e3,270)  
# reflections
r1=reflect(e1-e2,e1+2e2+e3)  
r2=reflect(r1,e1+2e2+e3)  
             
m=meet(e12,e13+2e23)  
             
a = e0+2e2-3e13
B = e12+2e13
# inverse : 1/a
inv1 (a)  
gp(a,inv1(a))  
inv1 (B) = (0,0,0,0,-1,0,-2,0) /5
gp(B,inv1(B))  
             
exp1 (e1-2e2)  
log1 (exp1 (e1-2e2)))  
# circular/hyperbolic functions
sin1 (0.2*e1+0.3*e2+0.5*e3)
asin1 (sin1(0.2*e1+0.3*e2+0.5*e3))
sqr=sqrt1 (e1-2e3)
pow (sqr, 2)
 
       
x = x1*e1 + x2*e2 + x3*e3
xx = gp(xx, x)
nabla(xx,x)
x1x2 = x1sin(x2)*e1 + x2cos(x1)*e2
nabla(x1x2,x)
 

nabla(nabla(x1x2,x),x)