### Applet: A three-dimensional linear transformation that reverses orientation

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The linear transformation $\vc{T}=B\vc{x}$, with $$B=\left[\begin{array}{rrr}3&1&-3\\1&3&-2\\1&1&-3\end{array}\right]$$ maps the unit cube to a parallelepiped of volume 14. The expansion of volume is reflected by the determinant $\det B = -14$. Since $\det B$ is negative, $\vc{T}$ not only expands volume by a factor of 14 but also reverse orientation, i.e., reflects objects into their mirror image. The reversal of orientation can be observed through the order of the colors on corresponding faces of the cube and parallelepiped. For example, the cube has a face with the colors ordered black, red, magenta, blue, when moving counterclockwise, while the parallelepiped has a face with the counterclockwise color order black, blue, magenta, red. You can further explore the mapping by changing either shape to other parallelepipeds by dragging the points on four of its vertices.

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The applet was made using three.js and requires Javascript as well as a browser that supports WebGL. For most three.js applets, you can drag with the mouse to rotate the view, drag with the right button to pan, and zoom in/out with the mouse wheel. Many applets contain points that you can drag to change values of variables.