# Geometric proof that cross product is linear

I've been searching online quite a bit for a simple and elegant geometric proof that the cross product distributes over addition. That is,

$$\mathbf a \times (\mathbf b + \mathbf c) = \mathbf a \times \mathbf b + \mathbf a \times \mathbf c.$$

I've found a few (such as this one and that one), but I feel these are all poorly written / presented and do not give me a good enough intuitive feel for why the cross product is linear.

Important: I do not seek a proof that derives from the formula for computing for the cross product based on vector components. I'm looking for a coordinate-free way of gaining intuition as to why the cross product distributes over addition this way using the intuitive "area of parallelogram and right-handed orientation" conceptual definition of the cross product.

Any thoughts, ideas, or insights?

we have $$\operatorname{area}(ABEC) = \vec{AB} \times \vec{AC}$$ and $$\operatorname{area}(BDFE) = \vec{BD} \times \vec{BE} = \vec{BD} \times \vec{AC}$$. Now \begin{align} (\vec{AB} + \vec{BD}) \times \vec{AC} &= \vec{AD} \times \vec{AC} \\ &= \operatorname{area}(ADFC) = \operatorname{area}(ABEC) + \operatorname{area}(BDFE) \\ &= \vec{AB}\times\vec{AC} + \vec{BD}\times\vec{AC} \end{align} This proves the additivity of the cross product.
The equality of the areas follows from the fact that \begin{align} \operatorname{area}(ADFC) &= \operatorname{area}(ADFC) - \operatorname{area}(ADB) + \operatorname{area}(CFE)\\ &= \operatorname{area}(ABEC) + \operatorname{area}(BDFE) \end{align}