for part a its the vector equation of the line (x, y, z) + lambda*(i,j,k) where (x,y,z) is the starting position vector and (i,j,k) is the direction vector and lambda is a scalar.
so say A is the starting position vector then the equation of the line is
(4, -3, 5) + lambda*(i,j,k)
(i,j,k) can be found by working out the vector needed for A->B
you know this line goes through C
then you basically have three equations as you know that all of them must satisfy the coords of point C.
So then you can work out lambda and then p.