Vector Cross Product Distributes over Addition/Proof 2

Proof
We draw a triangular prism whose parallel edges are in the direction of $\mathbf a$ and with its end faces as triangles with sides $\mathbf b$, $\mathbf c$ and $\mathbf b + \mathbf c$.


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

From Magnitude of Vector Cross Product equals Area of Parallelogram Contained by Vectors, the vector areas of these triangular end faces are $\dfrac {\mathbf b \times \mathbf c} 2$ and $\dfrac {\mathbf c \times \mathbf b} 2$.

The remaining vector areas are $\mathbf b \times \mathbf a$, $\mathbf c \times \mathbf a$ and $\mathbf a \times \paren {\mathbf b + \mathbf c}$.

From Total Vector Area of Polyhedron is Zero:
 * $\paren {\mathbf b \times \mathbf a} + \paren {\mathbf c \times \mathbf a} + \paren {\mathbf a \times \paren {\mathbf b + \mathbf c} } + \dfrac {\mathbf b \times \mathbf c} 2 + \dfrac {\mathbf c \times \mathbf b} 2 = 0$

from which we get:
 * $\paren {-\mathbf b \times \mathbf a} + \paren {-\mathbf c \times \mathbf a} = \paren {\mathbf a \times \paren {\mathbf b + \mathbf c} }$

The result follows from Vector Cross Product is Anticommutative:


 * $\mathbf a \times \mathbf b + \mathbf a \times \mathbf c = \mathbf a \times \paren {\mathbf b + \mathbf c}$