Vector Equation for a Line in R^3

This applet lets you investigate the vector equation for a line. The slider "t" changes the value of the scalar t and shows the position vector for the point that arises for the corresponding choice of t.
How do you parametrize the line segment from (1,2,2) to (3,-1,3)? That is, what values of t can you use in the vector equation for the line above to plot only the points on L between (1,2,2) and (3,-1,3)?