Rotation matrix construction

Question

I'm reading a book on how to construct transformation matrices and I'm stuck in a certain point.

From the book:

Now here's the figure that I don't understand:

How come the opposite edge in the right triangle is |Q|sin(theta)? When doing sine in that triangle I get opposite edge = |P'|sin(theta).

I don't understand the adjacent edge calculation as well, and equation (3.12) doesn't make sense to me as well.

Answer 1

The vector $Q$ is formed by rotating the vector $P$ through 90 degrees. Therefore $P$ and $Q$ have the same length.

The diagram is drawn very badly, in my opinion.

Answer 2

The author is trying to emphasize the point of view that the rotated vector $\mathbf P'$ is a linear combination of $\mathbf P$ and $\mathbf Q$. By construction, $\|\mathbf P\|=\|\mathbf Q\|$, so of course $\|\mathbf P\|\sin\theta=\|\mathbf Q\|\sin\theta$.