James Dean Palmer jdp5462 Week Number 1 1.2.17 http://http.tamu.edu/~jdp5462/math/1.2.17.html
Let x = (1,0) and y = (2,1). On a peice of graph paper plot and label the points tx+(1-t)y for t=-0.5, 0, 0.2, 0.5, 0.9, 1, and 1.2. From this example formulate a general principle. (What is special about the points corresponding to 0 <= t <= 1? These points are called convex combinations of the two given vectors.)
| t value | Intermediate Step | Result |
| -0.5 | (-.5,0) + (3,1.5) | (2.5,1.5) |
| 0 | (0,0) + (2,1) | (2,1) |
| 0.2 | (.2,0) + (1.6,.8) | (1.8,.8) |
| 0.5 | (.5,0) + (1,.5) | (1.5,.5) |
| 0.9 | (.9,0) + (.2,.1) | (1.1,.1) |
| 1 | (1,0) + (0,0) | (1,0) |
| 1.2 | (1.2, 0) + (-.4, -.2) | (.8,-.2) |
These results can be summarized in the following graph:
We can see from the intermediate step in the table above that for t<0 and t>1, the (1-t) y vector of the equation drives the result in the positive direction for negative t and in the negative direction for t > 1. For values of t between 0 and 1 x and y are both positive for both axis and thus complement each other as a convex combination; that is to say a straight line is formed between x and y for values of t between 0 and 1. The values that make up this line are the convex combinations of the two vectors.