I have a question that I think is meant to be bookwork..but I can't seem to find the proof anywhere and so have tried to work it out for myself.

I will write out the question and then the attempt at a proof of the first part.

Any help would be gratefully recieved!

I will denote *as powers and use ^ for the vector products to avoid confusion.

a) Suppose that a.b = 0

Find the solution r of the pair of equations:

r^a=b and r.c=n where a.c doesnt=0

Under what conditions does a solution exist if a.c=0? Find the most general solution in this case. Interpret your results geometrically.

b) Show that, if x doesnt=0 the equation r^d=xr+e has a unique solution and find it.

My attempt at a)

take vector product with a.

a^(r^a)=(a.a)r - (a.r)a=a^b

Hence r={1/|a|^2}a^b +ta where t = a.r / |a|*2

subsitute into r.c = n and we have

t =( n/a.c ) - [a,b,c]/(a.c)|a|*2

Therefore the solution is a single point.

I think that part is correct but I'm not sure, and have no idea how to continue with the problem! Please help!

Thank you!