CHAPTER 6: Question 7

2D explanations

To answer parts a-g, draw a point (p) on a set of Cartesian axes as shown below. Then, C21(z) means rotate by 180o about the z-axis. The effect of this transformation is to change the original x and y coordinates of the point to -x and -y respectively, but to leave the z coordinate unchanged. This is shown on the diagram below. Thus the new coordinates of the point are (-x, -y, z). Similarly, C21(y) moves the point to (-x, y, -z) and C21(x) moves the point to (x, -y, -z).

In a similar way to the above projections, reflection in a plane leaves two of the coordinates unchanged and changes the sign of the third. Thus s(xy) moves the point to (x, y, -z); s(xz) moves the point to (x, -y, z); and s(yz) moves the point to (-x, y, z). Finally, inversion about a centre of inversion at the origin will move the point from (x,y,z) to the origin, and then continue an equal distance on the other side of the origin so that the final coordinates are (-x, -y, -z). Projections for s(xy) and i are shown below

The answers to the two proofs are included in the simple answer.

 
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