Suppose that I have two arbitrary 3-dimensional vectors, $\vec{a}$ and $\vec{b}$. By the definition of the dot product, I can write
$$\vec{a} \cdot \vec{b} = \left|\vec{a}\right| \left|\vec{b}\right| \cos \theta$$
I can solve for $\cos \theta$:
$$\cos \theta = \frac{\vec{a} \cdot \vec{b}}{\left|\vec{a}\right| \left|\vec{b}\right|}$$
My question is, is it correct to rewrite $\cos \theta$ in terms of the unit vectors $\hat{a}$ and $\hat{b}$ as follows?
$$\cos \theta = \frac{\vec{a} \cdot \vec{b}}{\left|\vec{a}\right| \left|\vec{b}\right|}$$ $$\cos \theta = \frac{\vec{a}}{\left|\vec{a}\right|} \cdot \frac{\vec{b}}{\left|\vec{b}\right|}$$ $$\boxed{\cos \theta = \hat{a} \cdot \hat{b}}$$
where $\hat{a} = \frac{\vec{a}}{\left|\vec{a}\right|}$ and $\hat{b} = \frac{\vec{b}}{\left|\vec{b}\right|}$.
In other words, is it correct to reexpress the dot product in that way? Thanks for your time.
Yes, you can. You can get this result from your definition:
$$\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta$$
If you consider $\vec{a}$ and $\vec{b}$ to be unit vectors, then $|\vec{a}|=|\vec{b}|=1,$ and the expression becomes:
$$\vec{a}\cdot\vec{b}=\cos\theta,$$ where $\vec{a}$ and $\vec{b}$ are unit vectors.