- from P. Lounesto                                                                       ==>   Solutions
 
1.1   Let a = e2 - e12, b = e1 + e2, c = e0 + e2. Compute ab, ac. What is the instruction of this task ?  
1.2   Let a = e2 + e12, b = 1/2(e0 + e1). Compute ab, ba. Instruction ?
1.3   Let a = e0 + e1, b = -e0 + e1, c = e1 + e2. Compute ab, ba, ac, ca, bc and cb. Instruction ?
1.4   Let a = 1/2(e0 + e1), b = e1 + e12. Compute a², b²
1.5   Let a = e1 - 2e2, b = e1 + e2, r = 5e1 - e2. compute x, y in the decomposition r = xa + yb
1.6   Let a = 8e1 - e2, b = 2e1 + e2. Compute  a||  and  a_|_ .
1.7   Let r = 4e1 - 3e2, a = 3e1 - e2, b = 2e1 + e2. Reflect r first across a and then the result across b.
1.8   Let a = e0 + e1 + e12. Compute 1/a. Show that
       1/u = û  / (u û)       !=  (1/ (u û)) û
       1/u = (1 / (û u)) û  !=  û (1/ (û u))     with 1/(u) = cinv(u)   and   û = invol(u)  
       
       1/u = u~  / (u u~)       !=  (1/ (u u~)) u~
       1/u = (1 / (u~ u)) u~  !=  u~ (1/ (u~ u))     with 1/(u) = cinv(u)   and   u~ = rev(u)
1.9    Find the volume of the parallepiped with edges a = 2e1 - 3e2 +4e3, b = e1 + 2e2 - e3, c = 3e1 - e2 + 2e3
1.10  Find the inverse of the bivector B = 3e12 + e23
1.11  Let a = 2e1 + 3e2 + 7e3 and B = 4e12 + 5e13 - e23. Compute a ^ B and a _| B
1.12 Let a = 3e1 + 4e2 + 7e3 and B = 7e12 + e13. Compute the perpendicular and parallel components of a in the plane B
1.13  Show that the Clifford product of a bivector B and an arbitrary element u <- Cl3 can be decomposed as        B u = B _| u + B ^u + 1/2(B u - u B)    
1.14  Define the right contraction as u |_ v = ((e123 u) ^ v)/e123 for u, v <- Cl3
       Verify the folowing properties of the rigth contraction :
       x |_ y = x . y
       (u ^ v) |_ x = u ^ (v |_ x) + (u |_ x) ^ v'             (v' = invol(v))
       u |_ (v ^ w) = (u |_ v) ^ w
       for x, y  <-  R3 and      u, v, w  <-  ^R3
1.15  Show that u _| (v |_ w) = u |_ (v  |_ w)
1.16  Show that u ^ v - v ^ u  <- ^2 R3 and u v - v u <- R3 + ^2 R3
Let a <- R3, B <- ^2 R3, u = 1 + a + B
1.17  The exterior inverse of u  is 1 - a - B + alpha a ^ B with alpha <- R. Determine alpha.
1.18 The exterior square root  is  1 + a/2 + B/2 + beta a ^ B with beta <- R. Determine beta.
 
 













                        -  from W. Baylis
 
2.1 Consider the triangle of vectors c = a + b. Prove  a ^ b  = c ^ b  = a ^ c and show that magnitude of these products is twice the area of the triangle.
2.2 Let a, b, g be the interior angles of the triangle, opposite sides a, b, c respectively. Use the relation of outer product in the previous exercise to prove the law of sinus : sin a/a = sin b/b = sin g/c where a, b, c are the magnitude of a, b, c.
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