Consider the vector equation x = p + t(q - p) , where p and q correspond to distinct points P and Q in R2 or R3.
(a) Show that this equation describes the line segment PQ as t varies from 0 to 1.
(b) For which value of t is x the midpoint of PQ, and what is x in this case?
(c) Find the midpoint of PQ when P = (2, - 3) and Q = (0, 1).
(d) Find the midpoint o f P Q when P = (1, 0, 1) and Q = (4, 1, - 2).
(e) Find the two points that divide PQ in part (c) into three equal parts.
(f) Find the two points that divide PQ in part (d) into three equal parts.