The picture shows you what does to the standard coordinate grid. You see that sends the nice orthogonal (right angled) grid on the left to a non-orthogonal grid on the right.
Move the vector to see what does to other grids. Find a location for the vector that makes .
(Q5) Estimate the angle that makes with the -axis when .
Let be the angle when and let be the angle when .
(Q6) What's the exact sum of these two angles? .
(Q7) It turns out that for any matrix , . So for any , if , then and if , then < ___. So either way there must be a between and where .
The graphic also shows an orthogonal grid which remains orthogonal under the transformation. Having an orthogonal grid in the plot on the right makes it easier to solve .
Control the amounts of and to find an that sends to .
(Q8) The solution to is .
You can also solve by eye without moving .
On the right you see .
This tells you that the you need is .
Try to solve by eye.
(Q9) How much and add up to ? Answer: .
(Q10) The solution to is .
Verify your answers to the problems above by moving to .
Copyright © 2007 Todd Will
Created by Mathematica (April 15, 2007)