I have a difficult problem. A line is r=a+tb. A plane is r.n=p. Prove that they intersect at a+(p-a.n)b/(b.n). b.n is not equal to 0. p is a constant, b is a free vector parallel to the line.
Surely not that difficult! It's really just a matter of "doing what it says". At the point of intersection, on both line and plane, both of those equations are true. Replace the "r" in r.n= p with a+ tb and solve for t. Replace the t in r= a+ tb with that value of t.