I'm trying to work through the following theorem in Brannan et al.'s book "Geometry" (you might wish to skip ahead to the bold text "The Source of My Problem", so that you don't waste your time if the context is not necessary):

"A

matrix

represents a roation of

about the origin if and only if it satisfies the following two conditions:

(a)

is orthogonal;

(b)

."

So, the proof proceeds as follows:

A matrix

represents a rotation if and only if it is of the form

. (

*****)

I'm accustomed to, and am happy with, this fact (well, the 'if' part), and I am happy that it is easy to verify that if it is of this form, then

satisfies conditions (a) and (b).

Next, the proof procees to let

be a matrix that satisfies conditions (a) and (b).

Then, since

is orthgonal, the vector

has length 1; that is,

. Thus, there is a number

for which

and

.

Also, since

is orthogonal, the vectors

and

are orthogonal; that is,

(all fine by me so far) or