§ 5: answers

O17
i) λ(a₂b₃-a₃b₂,a₃b₁-a₁b₃,a₁b₂-a₂b₁), short: λa⊗b. Indeed, a⊗b is normal of the plane subtended by a and b.
ii) l⊗m. Indeed, l⊗m is direction vector of the intersecting line of two planes with normals through l and m, respectively.

O18
i) det(a,b,c)=0. Indeed, this is the condition that c belongs to the plane subtended by a and b.
ii) det(l,m,n)=0. Indeed, three planes through the origin go through one line if their normals lie in one plane.

O19 If det(a⊗a',b⊗b',c⊗c')=0 then det((a⊗b)⊗(a'⊗b'),(a⊗c)⊗(a'⊗c'),(b⊗c)⊗(b'⊗c'))=0.

O20 If det(a₁,a₂,a₃) = det(b₁,b₂,b₃) = 0, then det((a₁⊗b₂)⊗(a₂⊗b₁),(a₁⊗b₃)⊗(a₃⊗b₁), (a₂⊗b₃)⊗(a₃⊗b₂)) = 0.

O21 Start from λ(a,b,c) on l, this is λ(a,-2,1) or λ(1,0,0). The image point is λ(13-3a,16-4a,2a-8). This is the point λ(a,-2,1) if 3a-13 = a(8-2a). We get a =(5+√129)/4.