I have some difficulty interpreting this section of my lecture notes:
My question is with $(2.8)$.
How does $$\frac{cos \ \phi \ \vec i + sin \ \phi \ \vec j}{||cos \ \phi \vec i + sin \ \phi \ \vec j||}=cos \ \phi \ \vec i + sin \ \phi \ \vec j$$
hold true? I believe $||cos \ \phi \vec i + sin \ \phi \ \vec j||$ represents a number, as it is a magnitude. My thinking was, if it's a unit vector, that means the magnitude is $1$ and thus we have the numerator as the solution. But then, I run into an issue with:
$$\frac{-Rsin \ \phi \ \vec i + Rcos \ \phi \ \vec j}{||-Rsin \ \phi \ \vec i + Rcos \ \phi \ \vec j||}= -sin\ \phi \ \vec i + cos \ \phi \ \vec j$$
If the denominator is $1$, then the numerator and solution are not the same, which means my understanding is wrong. How did these right hand sides be reached?
Vectors $(\cos \theta, \sin \theta)$ are a bit special--they will always be magnitude 1, since we have the identity $(\cos\theta)^2+(\sin\theta)^2=1$.
If you take a vector of the form $\mathbf{x}=(R\cos \theta, R\sin \theta)$, the magnitude will be $R$, since $$|\mathbf{x}|=\sqrt{(R\cos\theta)^2+(R\sin\theta)^2}=\sqrt{R^2((\cos\theta)^2+(\sin\theta)^2)}=R\sqrt{1}=R.$$