Given vectors , and for scalars and :
- Plot the geometric vectors from to for the values
- If we let and run through all pairs of real numbers such that what is the set of points ? (Just give a graphical view, no proof necessary.)
- If we consider all pairs of points and with and make a sketch of the set of points we obtain.
- What is the set of all points if and ?
- What is the set of all points if we allow and to be any real numbers?
- The plot is as follows (with the locus of points for part (b) in red):
The points for each value of are
- The set of points as and run through real numbers such that forms a line. It is drawn in red in the plot in part (a).
- The set of points in the parallelogram formed by the origin, , , and .
- The horizontal band obtained by adding to the line for each .
- The whole plane .