Proof of equivalence of algebraic and geometric dot product? Geometrically the dot product of two vectors gives the angle between them (or the cosine of the angle to be precise). Algebraically, the dot product is a sum of products of the vector components between the two vectors.
However, both formulae look quite different but compute the same result. What is an easy/intuitive proof showing the equivalence between the two definitions?
I.e., demonstrate that $a_x b_x + a_y b_y + a_z b_z = \lvert a \rvert \lvert b \rvert \cos{\theta}$

TL;DR: So, in reality, while they do look different they actually do the exact same thing, thus outputting the same result. It's just that one is in regular coordinates and the other is in polar coordinates, that's all.

Let us consider the 2D case for simplicity.
Imagine we have a vector $\mathbf{r} = (x,y)$. We can represent this vector in polar coordinates as: 
\begin{align}
\mathbf{r} = (||\mathbf{r}||\cos(\theta), ||\mathbf{r}||\sin(\theta))
\end{align}
Consider two vectors $\mathbf{r_1}$, $\mathbf{r_2}$ with components $x_1$, $x_2$, $y_1$, $y_2$, $\theta_1$, $\theta_2$, then
$\mathbf{r_1} \cdot \mathbf{r_2} = (x_1, y_1) \cdot (x_2, y_2) = x_1x_2 + y_1 y_2$
and (in polar)
\begin{align}
\mathbf{r_1} \cdot \mathbf{r_2} &= (||\mathbf{r_1}||\cos(\theta_1), ||\mathbf{r_1}||\sin(\theta_1)) \cdot (||\mathbf{r_2}||\cos(\theta_2), ||\mathbf{r_2}||\sin(\theta_2))\\ 
&= (||\mathbf{r_1}||)(||\mathbf{r_2}||)  \cos(\theta_1) \cos(\theta_2) +  (||\mathbf{r_1}||)(||\mathbf{r_2}||) \sin(\theta_1) \sin(\theta_2)\\
&= (||\mathbf{r_1}||)(||\mathbf{r_2}||) [\cos(\theta_1) \cos(\theta_2) + \sin(\theta_1) \sin(\theta_2)]
\end{align}
Using the trigonmetric identity $\cos(\alpha - \beta) = \cos(\alpha) \cos(\beta) + sin(\alpha)  \sin(\beta)$ yields:
\begin{align}
\mathbf{r_1}
 \cdot \mathbf{r_2} = (||\mathbf{r_1}||)(||\mathbf{r_2}||) \cos(\theta_1 - \theta_2)
\end{align}
The difference in angle between both vectors is $\Delta \theta = \theta_1 - \theta_2 $. Therefore,
\begin{align}
\mathbf{r_1} \cdot \mathbf{r_2} = x_1x_2 + y_1y_2 = (||\mathbf{r_1}||)(||\mathbf{r_2}||)\cos(\Delta\theta)
\end{align}.
