Solution:
(a) No. A counterexample is: a = < 1, 0, 0 >, b = < 0, 1, 0 >, c = < 0, 0, 1 >

(b) No. A counterexample is: a = < 1, 1, 0 >, b = < b1, b2, 0 >, c = < c1, c2, >

b2 - b1 = c2 - c1 -> Some possible solutions are b = < 5, 2, 0 > and c = < 4, 1, 0 >

c) It follows that b = c:

First,

This means that a and b-c are perpendicular. Second

,

a and b-c are parallel.

This says that b - c is both parallel and perpendicular to the non-zero vector a.
Therefore b - c = 0, and b = c.