Dot Product Distributes over Addition/Proof 2

Proof
Let the vectors $\mathbf u$, $\mathbf v$ and $\mathbf w$ be embedded in a Cartesian $3$-space.

It is noted that $\mathbf u$, $\mathbf v$ and $\mathbf w$ are not necessarily coplanar.


 * Dot-product-distributes-over-addition.png

Let instances of $\mathbf u$ and $\mathbf w$ be selected so their initial points are at some point $O$.

Let an instance of $\mathbf v$ be selected so its initial point is positioned at the terminal point $U$ of $\mathbf u$.

Let the terminal point $\mathbf v$ be $V$.

Let $UA$ be dropped perpendicular to $\mathbf w$.

Let $VB$ be dropped perpendicular to $\mathbf w$.

Let another instance of $\mathbf w$ be selected so that its initial point is at $U$.

Let $VC$ be dropped perpendicular to this second instance of $\mathbf w$.

Let $CB$ be dropped from $C$ to the first instance of $\mathbf w$.

We have that:
 * $UA \parallel CB$

and:
 * $UC \parallel AB$

Thus $\Box ABCU$ is a parallelogram.

Hence:
 * $AB = UC$

Then we have that:

Hence the result.