I've done the first part of the attached question. 6lambda:1mu.
I don't think the hints make sense. c is perpendicular to a. How can it be on a plane with a and b and also be parallel to l as implied by r=a+kc?
c is not parallel to l. r = a + kc means that each point of l is a resultant of a and some vector collinear with c. This is guaranteed becase a and c form a basis for the plane that l lies in.
This is akin to the vector equation of a line with standard basis: r = b + kx.
I have consulted another source which was helpful and can confirm that you are wrong slider142.
I had been thinking that c is a position vector so I can understand it all better now. slider142 also seems to have thought that c is a position vector.
I calculated that lambda:mu is 6:1 but the book says 1:6. Is the book wrong? I think I can do all the rest of the question.