Chapter 2.4, Problem 80E

Consider the regular tetrahedron sketched below, whosecenter is at the origin.

Let T from ${ℝ}^3$ to ${ℝ}^3$ be the rotation about the axis through the points 0 and $P_2$ that transforms $P_1$ into $P_3$ . Find the images of the four corners of the tetrahedron under this transformation.
$\begin{array}{l}{P}_0\stackrel{T}{\to }\\ P_1\to P_3\\ P_2\to \\ P_3\to \end{array}$

Let L from ${ℝ}^3$ to ${ℝ}^3$ be the reflection about the plane through the points 0, $P_0$ , and $P_3$ . Find the images of the four corners of the tetrahedron under this transformation.
$\begin{array}{l}{P}_0\stackrel{L}{\to }\\ P_1\to \\ P_2\to \\ P_3\to \end{array}$
Describe the transformations in parts (a) through (c) geometrically.
a. $T^{-1}$   b. $L^{-1}$
c. $T^2=T\circ T$ (the composite of T with itself)
d. Find the images of the four corners under the transformations $T\circ L$ and $L\circ T$ . Are the two transformations the same?
$\begin{array}{l}{P}_0\stackrel{T\circ L}{\to }{P}_0\stackrel{L\circ T}{\to }\\ P_1\to P_1\to \\ P_2\to P_2\to \\ P_3\to P_3\to \end{array}$
e. Find the images of the four corners under the transformation $L\circ T\circ L$ . Describe this transformation geometrically.