Orthogonal Controls

Here's the picture of for another matrix .

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 .

Here's the picture of for another matrix .

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 .

Here's the picture of for another matrix .

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 .

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Copyright © 2007 Todd Will

Created by
Mathematica (April 15, 2007)