My textbook presented a proof of the distributivity of the dot product:
From the diagram, $\vert \mathbf{B} + \mathbf{C} \vert \cos\theta_3 = \vert \mathbf{B} \vert \cos \theta_1 + \vert \mathbf{C} \vert \cos \theta_2$. Multiply by $\vert \mathbf{A} \vert$. $\vert \mathbf{A} \vert \vert \mathbf{B} + \mathbf{C} \vert \cos \theta_3 = \vert \mathbf{A} \vert \vert \mathbf{B} \vert \cos \theta_1 + \vert \mathbf{A} \vert \vert \mathbf{C} \vert \cos \theta_2$. So: $\mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C}$ (Dot product is distributive)
I find this "proof" questionable. I'm struggling to come to terms with what the author did for this last part:
$\vert \mathbf{A} \vert \vert \mathbf{B} + \mathbf{C} \vert \cos \theta_3 = \vert \mathbf{A} \vert \vert \mathbf{B} \vert \cos \theta_1 + \vert \mathbf{A} \vert \vert \mathbf{C} \vert \cos \theta_2$. So: $\mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C}$ (Dot product is distributive)
This doesn't even seem like it's proving distributivity; rather, it seems like it's just applying the very property that it's supposed to prove. Am I correct, or is this actually a valid proof? Can someone please help me understand the "proof" step here? Thank you.
The proof is valid... sort of. There's a neglect of the obtuse case, but otherwise it's fine. The author is using the geometric definition: $$v \cdot w = |v| |w| \cos(\theta)$$ where $\theta \in [0, \pi]$ is the angle between $v$ and $w$.
With that in mind, they have let $\theta_1$ be the angle between $A$ and $B$, $\theta_2$ be the angle between $A$ and $C$, and $\theta_3$ be the angle between $B + C$ and $A$. The distributive law states $A \cdot (B + C) = A \cdot B + A \cdot C$, or in other words, $$|A| |B + C| \cos (\theta_3) = |A| |B| \cos(\theta_1) + |A| |C| \cos(\theta_2).$$ The above statement is what the author is trying to prove. They need to get a logical argument from a true statement that ends with the above equality, which is what they do. They begin with the observation that $$|B| \cos(\theta_1) + |C| \cos(\theta_2) = |B + C| \cos(\theta_3)$$ which is evident from the picture (there's some missing geometry stuff here, about how rectangles have pairs of parallel sides that are the same length, etc, but the picture itself is clear).
From this true statement, they multiply both sides by $|A|$ to get another true equality. As it so happens, this equality is the one they needed to end up with.
So yes, the proof is fine.