For each isometry:33 , we have an induced map:33that sends(,,)to(,,)(0,0,0). (General hint: for all the problems below, it will probably help to draw a picture of what happens in a particular example, and use geometric definitions of, for example, vector addition.)

1. Show that for any vector(,,), we have(,,)=(+,+,+)(,,).
2. Show that is a linear map on3and an isometry.
3. For each type of isometry of3(twist, etc.), show what type of isometry is.

For each isometry f: R3 > R3, we have an induced map L f: R3 > R3 that sends (a, b, c) to f (a, b, c) f (0, 0, 0). (General hint: for all the problems below, it will probably help to draw a picture of what happens in a particular example, and use geometric definitions of, for example, vector addition.) 1. Show that for any vector (x, y, z), we have Lf(a, b, c) = f(a + x, b + y, c + z) f(x, y, z). 2. Show that L f is a linear map on R3 and an isometry. 3. For each type of isometry of R3 (twist, etc.), show what type of isometry L f is